Synchronizing Global Supply Chains:
Advance Purchase Discounts
_______________
Wenjie TANG
Karan GIROTRA
2010/07/DS/TOM
Synchronizing Global Supply Chains:
Advance Purchase Discounts
by
Wenjie Tang*
and
Karan Girotra**
February 2010
*
PhD Candidate in Decision Sciences at IN SEAD, Boulevard de Constance, 77305
Fontainebleau Cedex, France, [email protected]
**
Assistant Professor of Technology and Operations Management at INSEAD, Boulevard de
Constance, 77305 Fontainebleau Cedex, France, [email protected]
A working paper in the INSEAD Working Paper Series is intended as a means whereby a fac ulty
researcher's thoughts and findings may be communic ated to interested readers. The paper should be
considered preliminary in nature and may require revision.
Printed at INSEAD, Fontainebleau, France. Kindly do not reproduce or circulate without permission.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE
DISCOUNTS
WENJIE TANG AND KARAN GIROTRA
Abstract. We study the economics of sharing demand information between a dual sourcing firm
and its network of retailers. Our analysis demonstrates that employing Advance Purchase Discount
(APD) contracts in these supply chains removes significant impediments to information sharing.
Essentially, these contracts synchronize the timeline of the dual sourcing firm’s decisions with the
actions of its network of retailers. This enables accurate, timely, and self-enforcing information
sharing, which reduces the demand–supply mismatches, and improves the profitability of each of
the agents in the supply chain. We provide prescriptions on the appropriate design of contracts
that enable this Pareto-improving information sharing. Next, we extend this analysis to incorporate
realistic constraints on the dual sourcing firm’s limited knowledge of its retailers’ operational costs
and information quality. We characterize “certainty-equivalent” values of the unknown retailer
parameters, which facilitate analogous prescriptions for the design of APD contracts in these realistic
settings. It is interesting that the unobservability of retailer parameters leads to an asymmetric
and “degree-of-unobservability dependent” departure from the full-observability design of the APD
contract. If the uncertainty in the unobserved parameter is small, then the optimal discount is higher
compared to the case of full observability; conversely, when the uncertainty is large, the optimal
discount is lower. It is significant that, despite the departure in the design of the APD contract, this
analysis reiterates that the benefits of APDs persist even under this practical constraint. Finally, we
report on the application of our prescription to a U.S.-based apparel wholesaler. Using data derived
from the operations of this firm, we estimate that APD-enabled improved information sharing can
increase net profits by about 17%.
February, 2010
1. Introduction
Over the last two decades, improved communication technologies, differences in input costs between
geographies, and geographical skill specialization have led firms to employ increasing levels of global
sourcing: sourcing products from locations far away from the market for the product (Hausman et al.
2005). The long distances between the sourcing locations and the market lead to long delivery lead
times and necessitate that firms make sourcing quantity decisions well before actual demand for
1
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WENJIE TANG AND KARAN GIROTRA
the product is realized. For firms sourcing short-life cycle products with uncertain demand, such as
fashion wear, this implies that the sourcing quantity decisions must be made with limited demandrelevant information. This leads to significant demand–supply mismatches and is a major drain on
firm profitability in these industries. Strategies to gather early, accurate demand information can
reduce these demand supply mismatches and have the potential to be a key competitive advantage
in this era of global sourcing (Donohue 2000, Taylor and Xiao 2009).
In addition to a firm’s best information about demand, its downstream partners in the supply chain,
who are closer to the customers, often have some additional demand-relevant information. Gathering this information in a timely fashion and using it to place more accurate orders at long–lead
time suppliers presents itself as an opportunity to reduce demand–supply mismatches. However,
traditional methods of acquiring this information—such as customer surveys, retailer shows, industry conventions, buyer events, and so forth—lead to poor accuracy and inappropriate timing of the
information. Specifically, the information shared by the partners is liable to be manipulated by the
communicating party to favor itself. Further more, the information cannot always be gathered at a
time of the firm’s choosing. It may be gathered too soon, in which case it will not reflect the latest
trends from the marketplace; or it may be gathered too late, in which case it cannot be used to
place orders at long–lead time suppliers.
This study proposes a contracting mechanism: an Advance Purchase Discount (APD) contract that
enhances the effectiveness of a firm’s global sourcing strategy by incentivizing accurate information communication from downstream tiers at a time chosen by the firm. Under such a scheme,
in addition to placing orders during the sales season, the firm offers its downstream retailers an
opportunity to place an additional order, in advance, at a discounted price. These early orders are
timed to coincide with the latest order time that ensures delivery from the long–lead time global
source. This advances some of the retailer’s demand and transfers some of the mismatch risk to the
retailers (Cachon 2004). More interesting and hitherto unidentified is the impact of this contract on
the information-sharing issues highlighted previously. Acting in their own best interests, the downstream retailers use the best information available to them in deciding the early order quantities.
From observing these orders, the upstream firm can infer the information available to the retailers
and can make its global sourcing decisions using the demand information inferred. Essentially, the
early order opportunity synchronizes the timeline of orders placed by retailers with the timeline of
decisions that the global sourcing firm must make. This alignment drives the timely communication
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
3
of accurate demand information by the downstream retailers to the firm. In the process, it also
redistributes risks in the supply chain in an efficient fashion. We show that, when taken together,
the timely communication of information and the redistribution of risk improve the supply chain’s
performance as a whole and can also improve the profits of both the retailers and the firm. We
demonstrate the robustness of this result by considering two different product designs: one where
products are customized for each market and one where the same generic product is sold in each
market.
Next, we examine the application of these contracts in a realistic setting. The contractual remedy
just described for inducing accurate and timely information sharing relies on designing incentives
that make information sharing preferable to any potential gains that the downstream retailers may
accrue by either manipulating or not sharing the information. As with most other contractual remedies, designing these incentives requires knowledge about the retailers’ benefits from manipulating
or not sharing the information. These benefits depend on the operational costs and quality of the
information available to the retailers. It is theoretically appealing to assume public knowledge of
these retailer side parameters, but this is unrealistic in most practical situations, and such assumptions often limit the actionable prescriptions from the contractual remedies proposed. To address
such concerns, we extend our proposal for employing APD contracts to this more practical setting
where the firm may have limited knowledge about its retailers’ operations. In this setup we characterize the optimal strategy in terms of a “certainty-equivalent” value of the unobserved parameter,
a value that the firm can use as if it knew the unobserved parameter for sure. It is interesting
that this value leads to an asymmetric and “degree-of-unobservability dependent” departure from
the full-observability design of the Advance Purchase Discount. When the uncertainty in the unobserved parameter is small, the optimal discount is higher than in the case of full observability; when
the uncertainty is large, the firm should set a lower discount. It is significant that this analysis
reiterates that the benefits of APDs—improving agents’ profits and the supply chain’s profits as a
whole—persist even under this practical constraint.
To validate the applicability of our analytical setup, to estimate the benefits accruing from our
prescriptions, and to develop practical guidelines for application, we close this study by reporting on
the implementation of our APD scheme at a New Jersey-based fashion-wear wholesaler. The subject
of our study, Costume Gallery Inc. (www.costumegallery.net) is one of the largest wholesalers of
dance costumes in the United States. The demand for dance costumes is highly seasonal, and almost
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WENJIE TANG AND KARAN GIROTRA
all of the annual demand is concentrated near the end of the school year in April. This demand
is highly influenced by cultural trends (this season, Michael Jackson costumes are exceptionally
popular). Costume Gallery sources extensively from Asia and has some local production, but it
sells the dance costumes through a network of U.S.-based dance schools (the retailers). These
dance schools typically buy costumes for dance productions they are organizing for their students.
Consequently, they have access to significant advance knowledge about the popular styles for the
season and the overall popularity of dancing as an activity. The supply chain structure, the sourcing
economics, and the informational environment in this setup are completely in line with our analytical
model. Further, the proposed APD contract was found to be simple and faced little resistance in
implementation. In particular, existing arrangements for managing accounts payables at most of the
affected firms were already employing similar terms for the purposes of cash flow management. Our
proposal was thus perceived to not depart significantly from the existing business culture. Using data
derived from Costume Gallery’s operations, we estimate that implementing our proposed scheme
would enable this firm to increase its net profits by about 17%. We close by reporting on some
practical issues encountered during the implementation.
The theoretical analysis demonstrating the effectiveness of our proposed contracting mechanism,
the case-based validation of its assumptions, and the enumeration of the accrued benefits reported
in this study make three important contributions to the theory and the practice of information
sharing in supply chains.
(1) We demonstrate that advance purchase discounts can be used in a supply chain to synchronize the timeline of different tiers of the supply chain; this induces accurate, timely, and
self-enforcing information sharing, which can make each of the agents in a supply chain
better off. We identify key drivers of these benefits and, based on this analysis, we develop
prescriptions for how a firm can go about designing and utilizing such a scheme to improve
its profitability.
(2) We extend our analysis to a realistic setting in which the firm has only limited knowledge
about downstream retailers’ operations. We illustrate analytically that all the benefits
highlighted previously persist in this more realistic setup, and we develop novel insights for
the design of APD contracts in these settings. We then report on the implementation of our
prescriptions at a real organization. This provides case-based evidence of the applicability of
our analysis and an estimate of the potential benefits derived from our prescription. Taken
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
5
together, the model extension and the study on implementation demonstrate the validity
of our analysis, improve the applicability of our prescriptions, and reinforce the utility of
advance purchase discounts in increasing firm profitability.
(3) Our illustration of advance purchase discounts as a Pareto-improving mechanism for information sharing brings together and enhances the largely parallel literatures on demand
information sharing and advance purchase discounts. Like this work, the existing demand information sharing literature has proposed contracts to achieve improved information sharing
(cf. Cachon and Lariviere 2001, Ozer and Wei 2006). We enhance this literature by considering an unexplored but widely prevalent dual sourcing arborescent supply chain structure, by
incorporating novel practical constraints on supply chain visibility and contract acceptance,
and by proposing a simple, optimized contract. The Advance Purchase Discounts literature
has studied these contracts as instruments for aligning incentives and sharing risks (Cachon
2004, Donohue 2000), and for incentivizing early consumer purchases (McCardle et al. 2004,
Tang et al. 2004). We enhance this literature by identifying a hitherto unrecognized benefit
of these discounts: credible, self-enforcing information sharing between different tiers of a
supply chain.
The rest of the paper is organized as follows. Section 2 reviews the parallel literatures on demand
information sharing and advance purchase discounts. Section 3 provides analytical results and
insights demonstrating the utility of advance purchase discounts. Section 4 illustrates the persistence
of these benefits even when there is limited knowledge about the downstream supply chain. Finally,
Section 5 reports on applying to a real organization the results from Section 3 and 4. We conclude
in Section 6.
2. Related Literature
This study brings together and enhances the largely parallel literatures on information sharing and
advance purchase discounts. We review our contributions to each of the two literatures in the
following two subsections.
2.1. Literature on Information Sharing. The information sharing literature typically studies a
supply chain in which a downstream agent has access to superior demand information that could
be used by an upstream agent to make better decisions regarding production, inventory, or pricing.
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WENJIE TANG AND KARAN GIROTRA
A key concern in this literature is the value of information sharing. Specifically, major findings are
that the value of information sharing is affected by coordination and information structures (Anand
and Mendelson 1997, Aviv 2001, 2002), production flexibility (Fisher et al. 1994, Fisher and Raman
1996, Cachon and Fisher 2000, Fisher et al. 2001, Milner and Kouvelis 2002), demand characteristics
(Lee et al. 2000, Milner and Kouvelis 2005), and competition and product characteristics (Ha et al.
2008, Ha and Tong 2008, Li 2002). Our analytical model follows this literature and models a similar
asymmetric informational environment. Lack of information sharing is also a prime driver of the
bullwhip effect (cf. Chen and Lee [2009] for a comprehensive treatment of this effect).
More recent work has identified impediments to realizing this potential value from information
sharing. While information sharing may create value for one agent in the supply chain, it might
not be beneficial to every other agent in the supply chain (Taylor 2006, Miyaoka and Hausman
2008). Further, information leakage might occur among competing agents (Anand and Goyal 2009).
Finally, the downstream agent may have the incentive to manipulate the information he obtains for
his own benefits, therefore impairing the truthful information sharing premise implied by the value
of information sharing literature.
Later developments in the information sharing literature provide remedies to these impediments. A
popular remedy is to design appropriate incentive and coordinating contracts so that, on one hand,
it’s in the best interest of the downstream agents to truthfully share their private information and
on the other hand, the truthful information makes the supply chain better off. Our study continues
in this tradition but achieves this remedy in our dual sourcing context by employing a simple, intuitively appealing, and novel contract—the Advance Purchase Discount Contract. Previous work has
identified different contract types that achieve the same goal in other contexts; these include return
and rebate policies (Arya and Mittendorf 2004, Taylor and Xiao 2009), contracts with commitment
and options (Cachon and Lariviere 2001, Ozer and Wei 2006), and wholesale price contract under
confidentiality (Li and Zhang 2008) and with customer review strategies (Ren et al. 2008).
Our remedy to enable information sharing distinguishes itself from the previously proposed ones in
terms of the applicable context, its robustness to realistic constraints, in the simplicity and intuitiveness of the proposed contract, and in terms of optimally incentivizing self-enforcing information
sharing.
Our applicable context is unique in terms of the supply chain structure and the nature of the
supply chain relationships. Our contract is designed to enable information sharing between a dual
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
7
sourcing firm and its network of retailers. We explicitly model and consider the unique economics
of this widely prevalent supply chain structure. Specifically, this implies that unlike the stylized
economic models in Arya and Mittendorf [2004] and Li and Zhang [2008] which abstract from the
internal physical flows in the supply chain, our operational analysis models individual agents in the
supply chain such that they are allowed to carry physical inventories and face the risk of over- and
under-supply. Further, unlike the single-source capacity reservation model of Ozer and Wei [2006],
our study applies to products where shortage of production capacity is not a significant concern.
Finally, in contrast to Cachon and Lariviere [2001], in our context, it is the upstreamagent who
needs the information offers a contract to infer private information from the downstream agent.
The supply chain relationships considered in our model depart from those studied in Ren et al. [2008],
where the relationships are characterized by long-term commitments to enable inter-temporal tradeoffs which significantly alter the enforcement incentives. Our proposed remedy applies equally even
in the absence of long-term commitments.
In terms of the specifics of our contract design, unlike Ozer and Wei [2006] who also consider
a time sensitive discount contract in a restricted capacity context, we endogenize a previously
exogenous contract parameter—the discount in the APD contract, to optimally incentivize selfenforcing information sharing and create just the right incentives for each participating agent.
Our work also departs from the exitsing information sharing literature in incorporating two new
practical constraints: First, our extended model explicitly considers a major practical constraint that
has limited the implementation of most contracts—the contract designing agent’s limited knowledge
about the cost structure and quality of information available to the opposing agent. All existing
literature assumes full knowledge of these parameters, which in an information asymmetry context,
is conflicting with the philosophical purpose of the contract itself. Our extended model considers
this limited visibility and provides actionable prescriptions and appropriate contract designs for
these contexts too. To the best of our knowledge, we are the first to consider such lack of visibility
in the information sharing literature. Secondly, the contract type proposed in our model is simple
and intuitive—getting a discount for timely actions and is very often used by businesses in the
context of account payables; therefore it is easier to introduce it in within the constraints of the
existing business culture.
Finally, we also report on the practical application of our contract in a real organization. To the
best of our knowledge, this is the first paper that reports on a practical application of an information
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WENJIE TANG AND KARAN GIROTRA
sharing contract and provides some data-based estimates of the benefits of employing an information
sharing contract.
To summarize, our work departs from the extensive information sharing literature by considering a
novel but widely prevalent dual sourcing arborescent supply chain structure and a broad spectrum
of supply chain relationships, by incorporating practical constraints on supply chain visibility and
contract acceptance, and by proposing an optimized simple contract.
2.2. Advance Purchase Discount Literature. Our work is also related to the Advance Purchase
Discounts literature. Advance purchase discounts were first studied as direct offers from a retailer to
consumers in the market: Tang et al. [2004] consider an “advance booking discount” program that
entices consumers to commit to their orders at a discounted price prior to the selling season, which
shifts some of the demand earlier and enables retailers to better manage inventory and capacity.
McCardle et al. [2004] further develop these consumer advance purchase discount contracts to a setup
that includes retailer competition. Xie and Shugan [2001], Dana [1998], Gundepudi et al. [2001], and
Akan et al. [2009] also consider consumer advance purchase discount contracts and establish their
utility as a price discrimination mechanism. Li and Zhang [2009] study the interaction of the above
two benefits of an APD contract: advance demand information and price discrimination. More
relevant to our study is the literature on APDs offered in a supply chain. While the APD contract
offered to agents in the supply chain is similar to consumer APD contracts, agents in a supply chain
are typically assumed to have different utility and incentive structures from those of consumers,
which leads to significantly different analysis and implications. In particular, new issues of risk
distribution and incentive coordination that are not relevant in consumer APDs play an important
role in these settings. The pioneering works by Donohue [2000], Cachon [2004], and Ozer et al.
[2007] were the first to identify these effects. However, neither Donohue [2000] nor Cachon [2004]
considers the information asymmetry between different tiers of the supply chain. Ozer et al. [2007]
does consider information asymmetry, but the private information possessed by the downstream
agent is not shared within the supply chain. More recently, Bernstein et al. [2009] apply these
contracts to a supply chain with multiple locations and transhipment, while Cho and Tang [2009]
study market price risks along with the usual demand–supply risks.
Finally, our work also extends traditional information-sharing and contracting models to capture an
additional type of practically relevant asymmetric information—namely, asymmetric information
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
9
that arises from limited knowledge about the costs and information precision of the supply chain
partners. This asymmetry has been previously studied in isolation with other information-sharing
issues (cf. Kayis et al. [2007]) and our work integrates these two approaches. There is a growing
body of literature that studies supply chain management with limited information about the environment (cf. Perakis and Roels [2008]). Our extension with asymmetric information continues in
this tradition.
3. Benefits of Advance Purchase Discounts: Customized Products
In this section, we develop a stylized model in order to illustrate the benefits of employing Advance
Purchase Discount (APD) contracts, to understand the key drivers of these benefits, and to identify
some challenges in employing these contracts.
Our model builds on the basic setup described in the celebrated case on Sport Obermeyer (Fisher
et al. 1994, Hammond and Raman 1994, Fisher and Raman 1996, Parlar and Weng 1997, Donohue
2000), where the procurement of a short–life cycle product must be allocated between a “speculative”
production source with long lead times and a “reactive” production source with short lead times.
The reactive production source can make use of the latest demand information to meet any shortfalls
from the speculative source. Our model replicates the speculative–reactive sourcing choice of a firm
like Sport Obermeyer, yet we add three interesting features to this much-used model as follows.
(i) In addition to modeling the upstream sourcing choices of the firm, we also explicitly model
the actions and strategic behavior of the firm’s downstream supply chain—that is, we model the
network of retailers through which the dual sourcing firm sells its products. (ii) We explicitly
model the information environment in which the firm and its retailers make these choices, including
allowances for any private information that may be available to the retailers and the consequent
information asymmetry. (iii) We analyze and compare alternate contracts between a firm similar
to Sport Obermeyer and its downstream retail network.
3.1. Supply Chain Setup and Information Environment. Consider a firm that sells a short–
life cycle product through a network of N geographically dispersed retailers (Figure 1). Products
with seasonal demand (e.g., fashion wear, toys, seasonal sports goods), rapidly evolving consumer
electronics, and holiday decorations all fit this setup. In addition, perishable products that fit
the classic newsvendor setup also fit. The retailers source the product from the firm with near
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WENJIE TANG AND KARAN GIROTRA
Figure 1. Setup: The Supply Chain
instantaneous lead times and sell it in the market for a unit price, Ri . We denote the selling season
demand from each of the multiple retailers as Di , where i ∈ {1, 2, ..., N }.
In this section, we build a model where this product must be customized for each of the different
retailers or territories. This setup applies to products sold as private-label, store-branded products, or products that must be customized for different regulatory standards in different territories.
We further assume that it is not efficient to employ strategies involving delayed differentiation or
postponement; in other words, the product design may not be modular or strategic, and marketing considerations limit the effectiveness of postponement strategies (Anand and Girotra 2007).
Consequently, the product must be procured and inventoried separately for each of the different
retailers/territories.
In Section 7 we present an alternate model where the same generic product is sold through all
retailers under identical contract terms, thus enabling further efficiencies from joint production and
common inventories. The analysis and the dynamics of the alternate model are somewhat different
than the model of this section, but each of the key results presented here holds also in the case with
generic products.
Demand in each of the nonoverlapping markets is independent and distributed normally with mean
µi and variance σi2 = 1/ρi . Each retailer has access to some intelligence about the local market
conditions, private information that we model as a signal of demand, Yi . Following Li and Zhang
[2008], we assume that the signal is distributed conditionally normal, or that Yi |Di is distributed
normally with mean Di and finite variance 1/ti . The signal is thus an unbiased estimate of the
true demand, where ti can be interpreted as the precision or quality of the retailer’s local market
knowledge. Further, we assume that all agents utilize information signals in a Bayesian fashion
(see Winkler 1981). Consequently, retailers, having access to this private information signal can
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
11
Figure 2. Wholesale Price Contracts: Sequence of Events
update the common prior on the demand distribution with the private information signal to obtain
0
the private posterior distribution of demand, Di |Yi , which is distributed normally with mean µi =
0
(ρi µi + ti Yi ) / (ρi + ti ) and variance σi2 = 1/ (ρi + ti ).
The firm can produce or source these products using multiple different sources, each with different
delivery lead time and associated costs. To simplify our analysis, we will consider two extreme
alternate sources for this product: the product can be sourced at unit cost CL from a local source
with nearly instantaneous lead times. As an alternative, the product can be sourced at unit cost
(1 − γ) CL , where γ ∈ (0, 1), from an overseas source with a significantly longer lead time. In particular, orders to the overseas source must be placed well ahead of the selling season of the product,
which necessitates a make-to-stock inventory strategy. In contrast, a make-to-order strategy may
be employed with respect to the local source. All of the described parameters, except the private
information signal Yi , are assumed to be common knowledge to all agents.1
3.2. Wholesale Price Contracts: Sequence of Events and Performance. We first examine
the conventional supply chain practice of using wholesale price contracts; this will enable us to
develop a benchmark for comparison with an alternate contracting form, the Advance Purchase
Discount (APD) scheme to be described in Section 3.3. The sequence of events is illustrated in
Figure 2. First, well ahead of the selling season, for each retailer i, the firm places an order of P0i
customized units from the overseas source at unit cost (1 − γ) CL . During the selling season, after
demand is known, the retailers order the realized demand from the firm at unit price wi .2 Next,
1In line with Taylor and Xiao [2008] and with most existing work on supply chain contracting, we assume that the
firm has full knowledge of retailer-side operational parameters. In particular, the retailer’s signal quality ti and the
retailer’s ordering costs are all discerned from past experience. To the extent that this past experience is sufficient, the
assumption that the retailer’s signal quality is common knowledge is appropriate. However, our own experience with
applying the results of this study did not show this to be the case. Nevertheless, we first build the basic intuitions
under this assumption. Then in Section 4, we relax this assumption to capture a more realistic settting in which the
firm is not fully aware of all relevant retailer-side parameters and can only speculate on their distribution.
2Setting the wholesale price is often influenced and constrained by various aspects of the firm’s market strategy,
competitive landscape, etc. Within the operations perspective of our paper, we assume that the wholesale price is
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WENJIE TANG AND KARAN GIROTRA
the firm examines its inventories; if its3 inventory is not sufficient to meet demand from any of the
retailers, the firm procures additional units for that retailer from the local source at unit cost CL .
Finally, each retailer’s demand is fully met by supply from the overseas sources and, in some cases,
extra production from the local source. Any items left over from the initial orders at the overseas
source are worthless to the firm.
In this setup, the retailers place their orders after market demand is realized and thus need not
maintain any inventories or face any inventory risk. Each retailer’s expected profits are calculated
as her margin multiplied by the demand mean. Given the customization necessary for each product,
the firm needs to maintain separate inventories for each of the retailers. Each of these setups is
individually a classic newsvendor setup with recourse to a reactive capacity source, the local source.
Consequently, the firm’s overseas order quantities are P0i = µi + σi zγ for all i, and the consequent
P
−1 (γ) and where φ (·)
expected profits are N
i=1 [(wi − (1 − γ) CL ) µi − CL φ (zγ ) σi ], where zγ = Φ
and Φ (·) are the standard normal pdf and cdf respectively.
3.3. The Advance Purchase Discount Scheme. We now propose an alternate contracting form
between the firm and its network of retailers, the Advance Purchase Discount (APD) scheme. The
setup of the market, the retailers, the firm, the informational environment, and the economics of
sourcing from the local and global source are all identical to the wholesale price contract described
previously. However, unlike the wholesale price contract, the firm now offers each retailer i an
opportunity to place an order in advance of the selling season at a discounted price of (1 − δi )wi ,
where δi ∈ [0, 1]. The retailer may choose to purchase a certain quantity Qi during the preseason
at this discounted price. The retailer’s tangible administrative cost incurred as a result of placing
this additional order is captured by Ki , where 0 ≤ Ki ≤ wi µi .4 This administrative cost may
include financing costs, order costs, retailer investments in forecasting, and all other tangible fixed
costs associated with an additional preseason order. During the selling season and after the demand
uncertainty is resolved, each retailer can order additional units at a price wi . From the firm’s
perspective, aside from the APD scheme offered, the operations remain identical to the benchmark
wholesale price contract case.
exogenously given, and instead we focus on the examination of novel operational practices that improve the Supply
Chain’s performance for each given value of wholesale price.
3Throughout the paper, we use “it” in reference to the firm, and “she” in reference to any retailer.
4In order to avoid trivial solutions where the administrative cost is extremely prohibitive, we assume that the administrative cost is bounded—in particular, it is lower than the total full-price order-cost for the retailer.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
13
Figure 3. The Advance Purchase Discount Scheme: Sequence of Events
The sequence of events now proceeds as in Figure 3. As before, the retailers have access to their
private demand signals. In advance of the selling season, the firm proposes the APD contract with
discount δi . Next, the retailer decides if she wants to participate in the APD scheme. If she decides
to participate in the scheme, the retailer updates the common prior on the demand distribution with
the private signal to obtain the posterior distribution of demand, Di |Yi and orders Qi units at price
(1 − δi ) wi . For each retailer i, the firm then places its orders Pi from the overseas source at a unit
cost of (1 − γ) CL . After demand Di is realized, the retailers place additional orders for (Di − Qi )+
units at price wi . Next, the firm places orders (Di − Pi )+ from the local source when the overseas
orders cannot satisfy the demand from retailer i. Finally, all deliveries and payments are made and
profits are realized.5 This setup differs from the previously described benchmark wholesale price
contract setup in only one way: the Advance Purchase Discount, which gives retailers the ability to
place preseason orders at a discounted price.
Similar schemes have previously been proposed in two different contexts in the existing literature.
In contrast to our setup, where the scheme is offered to different retailers within a supply chain, the
APD scheme is offered to a market’s atomistic heterogeneous customers in Dana [1998], Gundepudi
et al. [2001], and Tang et al. [2004]. These customers have different incentive structures than our
retailers, and our analysis will therefore illustrate some novel dynamics. Cachon [2004] and Donohue
[2000] also introduce a similar scheme. As in our setup, their scheme is offered by an upstream firm
to its retailers, but in the absence of demand information asymmetry, leading to a different analysis.
5Observe that in this setup the retailers cannot return “over-ordered” units to the firm. We make this assumption
for two reasons. First, return policies are typically adopted to mitigate retail risk by incentivizing retailers to order
the supply chain optimal quantities. However, given the APD contract and our two order modes, the supply chain is
always coordinated regardless of retailers’ order quantities. Therefore, a return policy does not help coordinate the
supply chain. The second reason is that, owing to the nature of customized products, the returned units are useless
to the firm; hence, the firm also prefers to avoid such a policy. We relax this assumption in Section 7 to examine the
case of generic products, where it may be in the firm’s interest to allow the return of extra units from retailers’ early
orders when those units could be used to supply other retailers.
14
WENJIE TANG AND KARAN GIROTRA
The focus of these two papers is on supply chain coordination. The two-order mode of our setup—
retailers can also order after demand is realized to compensate for insufficient early orders—captures
the demand characteristics of seasonal products and also eliminates concerns about a loss of supply
chain efficiency. That is, in employing either a wholesale price contract or an APD contract, there
is full coordination in the firm-retailer supply chain. This allows us to focus on the informationsharing issues central to our study and to abstract away from the previously examined supply chain
coordination issues. We next provide an analysis of the equilibrium strategies of the retailers and
the firm under this Advance Purchase Discount scheme.
3.4. Strategies under Advance Purchase Discounts. The setup described so far is a multistage game of imperfect information, where the private information Yi of each retailer defines the
type space. We describe the Bayesian subgame perfect strategies in this game.
The strategies and equilibrium actions for decisions made after the realization (by the firm and its
retailers) of demand follow directly from the setup in Section 3.3. Essentially, each retailer orders
the required quantity, if any, from the firm to meet all available demand. Similarly, the firm meets
its market-specific shortfall by procuring from the local source. The preseason stage of the game is
more interesting. In this stage, the firm must decide on the specific APD parameters to offer and
the retailers must choose their advance purchase quantities, if any, on the basis of these parameters.
The following lemma characterizes the retailers’ strategy in response to the firm’s offer.
Lemma. Retailers’ Strategy Profile: Each retailer orders an advance purchase quantity Qi and
earns expected profits EπiR , where
0
(3.1)
0
0
Qi = µi + σi zδi , EπiR = (Ri − (1 − δi ) wi ) µi − wi φ (zδi ) σi − Ki if δi ≥ δ i ,
Qi = 0,
(3.2)
EπiR = (Ri − wi ) µi
otherwise,
n o
0
δ i = δi δi wi µi − wi φ (zδi ) σi = Ki ,
and zδi = Φ−1 (δi ).
Proof. All proofs are provided in the Appendix.
The retailer’s strategy profile stipulates her order in the preseason for any given APD contract offered
by the firm. The retailer’s strategy can be viewed as consisting of two decisions: first, whether the
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
15
retailer should participate in the APD scheme, thereby incurring an additional administrative cost;
and second, how much should the retailer order if she participates.
We demonstrate that there exists a δ i such that, at this discount, retailer i is indifferent between
participating in the APD scheme or not—that is, the expected profits from participating in the APD
scheme equal those from not participating. Moreover, for any discount greater than δ i , retailer i is
willing to participate but otherwise is not. We call this threshold level δ i , the retailer’s minimum
acceptable discount (MAD).
More specifically, this threshold level is characterized by the discount at which the benefits from
the APD scheme are equal to the costs (Equation 3.2). The benefits of the APD scheme to the
retailer arise from sourcing some fraction of the product at a discounted price, which is captured
by the term δi wi µi . There are two sources of costs from participating in the APD scheme. First,
the retailer now places orders in advance of the demand realization and consequently bears some
0
demand–supply mismatch costs; these are captured by the term wi φ(zδi )σi . Second, the retailer
now incurs the fixed administrative cost Ki of an additional preseason order.
The retailer compares the benefits and the costs of participating in the discount scheme. If the
preseason discount offered is deeper than the threshold discount, then the savings due to sourcing at
the discounted price are more than the mismatch and administrative costs and so the retailer chooses
to participate in the APD scheme. In contrast with supply chain APD schemes previously considered
in literature (e.g., Cachon 2004), we endogenize and explicitly model the retailer’s decision to
participate in the APD scheme. This leads to the novel dynamics just described.
Once the retailer decides to participate in the APD scheme, she obtains the private information
and faces a newsvendor-like choice, with consequences from ordering too much (penalty of leftover
inventories) or from ordering too little (penalty of sourcing later at full price rather than the discounted price). Note that if the retailer decides to participate in the APD scheme, she chooses
her order quantity on the basis of all the information available to her. In particular, she uses the
private demand information signal to update the common knowledge prior demand and uses the
consequent posterior estimates on demand. This leads to an expression for Qi in Equation (3.1),
that is a function of the posterior estimates.
0
Recall that µi , the posterior estimate of the mean of the demand distribution, is a function of Yi ,
we remark that the retailer’s order quantity Qi is a fully invertible function of the private signal
16
WENJIE TANG AND KARAN GIROTRA
Yi . Thus, by observing the retailer’s order, the firm can fully discern the retailer’s private signal.
This and previous observations suggest that, if a firm so desires, it can offer a discount substantial
enough to induce a retailer’s participation in the APD scheme and thereby perfectly infer her private
information!
Next, we examine the firm’s strategy before outlining the equilibrium outcome.
Lemma. Firm’s Strategy Profile: The firm offers each retailer a discount δi and places an
overseas order Pi for each product,
0
0
δi = δ i , Pi = µi + σi zγ
if Ki ≤ K̄i ,
δi = 0,
otherwise,
P i = µi + σ i zγ
0
where K̄i = CL φ(zγ )(σi − σi ). The firm’s expected profits Eπ F are
F
i=1,...,N Eπi ,
P

 ((1 − δi ) wi − (1 − γ) CL ) µi + (wi φ (zδ ) − CL φ (zγ )) σ 0
i
i
F
Eπi =
 (w − (1 − γ) C ) µ − C φ (z ) σ
i
L
i
L
γ
i
where
if Ki ≤ K̄i ,
otherwise.
The firm’s preseason strategy consists of two key decisions: what discount δi to offer, and how many
units Pi to order from the overseas source. These decisions must be made while bearing in mind the
retailer’s reaction to the contract offered. The discount to offer determines the retailer’s decision
to participate or not. It is thus driven by two factors: the benefits and the costs of inducing the
retailer to participate.
Once the retailer participates in the APD scheme, the firm can infer her private information. This
allows the firm to update its demand forecast; in particular, the demand forecast now has a lower
variance. Hence the firm can place its overseas orders on the basis of this updated, lower-variance
demand forecast, which reduces the firm’s demand–supply mismatch costs. Specifically, those costs
0
are reduced by CL φ(zγ )(σi − σi ). The first part of this expression, CL φ(zγ ), captures the cost
differential between the overseas and local sources, which drives the mismatch costs. The next part,
0
(σi − σi ), captures the reduction in variance from obtaining the retailer’s private information. In
addition to reducing the effective mismatch costs, the firm also transfers an additional amount of
these costs to the retailer. In particular, since the retailer now carries inventories, she also bears
some of the oversupply risk. These two components constitute the firm’s benefits from retailer
participation.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
17
From the firm’s point of view, there are also costs associated with incentivizing the retailer to
participate. The firm must offer a discount deep enough that the retailer’s benefits from the discount
exceed the sum of her mismatch costs and her administrative cost Ki .
Considering both the benefits and costs of inducing retailer participation, we can see that if the
administrative cost for the retailer is small, then the firm’s cost to incentivize her participation
is smaller and so the firm finds that the potential benefits from offering APD outweigh the costs.
Specifically, we show that there exists a threshold administrative cost K̄i such that, if the administrative cost is lower than this threshold, the firm finds it profitable to incentivize the retailer to
participate. A firm that finds this profitable will set the discount just deep enough to make the retailer indifferent between participating and not. As discussed in the previous lemma, this threshold
discount is the retailer’s MAD, δ i .
The decision regarding the overseas order quantity follows in a relatively straightforward fashion.
The firm is now facing a newsvendor situation, albeit with demand forecasts updated on the basis
of information discerned from observing the retailer’s order.
3.5. Benefits of Advance Purchase Discounts. In this section we examine the effects of deploying an Advance Purchase Discount scheme. The next theorem characterizes conditions and a
class of APD contracts that make the retailers, the firm, and the supply chain as a whole better off.
Theorem 1. For all i such that Ki < K̄i , there exists an advance purchase discount δi ∈ δ i , δ¯i
such that retailer i, the firm, and the supply chain are all strictly better off. The benefits are given
as
0
∆EπiR = 1Ki <K̄i δi wi µi − wi φ (zδi ) σi − Ki > 0,
∆Eπ F =
N
X
0
1Ki <K̄i −δi wi µi + wi φ (zδi ) σi + K̄i > 0,
i=1
∆EπiS =
N
X
1Ki <K̄i K̄i − Ki > 0,
i=1
n o
0
where δ̄i = δi δi wi µi − wi φ (zδi ) σi = K̄i and where δ i and K̄i are defined in the preceding lemmas.
18
WENJIE TANG AND KARAN GIROTRA
This theorem demonstrates the existence of a set of feasible discounts that make the retailers and
the firm strictly better off. Essentially, these benefits accrue owing to a combination of the following
four effects, each arising out of offering the Advance Purchase Discount.
(1) Effect Admin: To avail the APD, the retailers must process an additional preseason order,
which leads to an additional administrative cost of Ki .
(2) Effect Discount: By availing the APD, the average transfer price between the firm and
the retailers is reduced; this leads to higher expected profits for the retailers but to lower
ones for the firm. This effect is captured by the term δi wi µi .
(3) Effect Information: In availing the APD, the retailers signal to the firm their private
information about demand. Hence, the firm can utilize this information to update its
demand forecasts and reduce their variance. Specifically, for demand from each retailer i,
0
the firm’s forecast variance is reduced to σi2 from the prior demand variance σi2 . This allows
the firm to better match its overseas order with the market demand and thereby reduce its
0
demand–supply mismatch costs, as captured by the term CL φ(zγ )(σi −σi ). An even stronger
0
effect is that σi is lower for retailers with better private information. Effect Information is
novel in our setting and captures the role of APD in promoting better forecast sharing in
supply chains. The literature on consumer-centric APD schemes includes a conceptually
similar effect (cf. Tang et al. 2004), but it does not have the interesting interaction with our
next effect on demand–supply mismatch risks.
(4) Effect Risk: In availing the APD, the retailers end up bearing some new demand–supply
mismatch risk. In the absence of an APD, the retailers did not stock any inventory. After
taking advantage of the APD, retailers generally have leftover inventories, and this reduces
their expected profits. On the other hand, by the same mechanism the firm ends up transferring some of this oversupply risk to the retailers, reducing its oversupply risk by the same
0
amount. This is captured by the term wi φ (zδi ) σi . A similar risk transfer effect is captured
by the model in Cachon [2004], but its lack of information asymmetry means that there is
no interplay with Effect Information.
The interplay of these effects drives the net advantage (or disadvantage) of an APD scheme for
each of the relevant agents. Let’s first consider the retailers. Effects Admin and Risk reduce the
expected profits of the retailers, and Effect Discount increases their expected profits. The expression
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
19
¯
for ∆EΠR
i captures the trade-off between these effects. For the range of discounts δi ∈ δ i , δi , the
net effect is that the retailers are better off by participating in the APD scheme.
Next consider the firm: Effect Discount reduces its average sales price and its expected profits;
whereas Effect Information and Risk increase its expected profits. For each retailer that the firm
faces with Ki < K̄i , there exists a discount at which the net effect for the firm is positive. Note
that the firm cannot actually accrue these benefits unless the retailer has nonnegative benefits of
participating in the scheme; although the firm proposes the scheme, it is the retailers who actually
participate in it and generate the benefits.
Finally, consider the supply chain as a whole. Effects Discount and Risk are internal effects: they
involve transfers between the firm and retailer, and do not affect the supply chain as a whole.
The impact of APDs on the supply chain is composed of (i) the benefits due to better transfer of
information from the tier (the retailers) that has access to it to the tier (the firm) that makes a
crucial supply chain–relevant decision and (ii) the loss due to the additional administrative costs.
By definition of K̄i , the benefit of the information sharing is higher than the administrative cost
when Ki < K̄i .
Theorem 1 shows that, if the administrative cost of an additional order is not excessive, then the
supply chain as a whole gains by better information sharing. This highlights the previously overlooked advantage of advance purchase discount as a self-enforcing information-sharing mechanism.
The total benefits accruing to the supply chain by virtue of this superior information sharing can
now be divided between the firm and the retailers to make everyone better off. In particular, the
extent of the discount, δi , determines what fraction of the total gain accrues to the firm and what
fraction to the retailer. If δi = δ̄i then all benefits accrue to the retailer; on the other hand, if
δi = δ i then all benefits accrue to the firm. In our setup, the firm moves first and proposes a
take-it-or-leave-it APD contract, so it enjoys all the bargaining power. Not surprisingly, the firm
uses this power to ensure that it retains all the benefits from the APD; the firm does this by setting
the discount to be just substantial enough that it ensures the retailers’ participation but does not
share any benefits beyond that. The next section illustrates this intuition.
3.6. Equilibrium Outcome. The equilibrium outcome follows from the strategy profiles for the
retailers and the firm, which were described in Section 3.4. If Ki ≤ K̄i , then it is in the firm’s
interest to offer the Advance Purchase Discount and it is in the retailer’s interest to purchase some
20
WENJIE TANG AND KARAN GIROTRA
quantities in advance. This purchase serves as a signal of the information available to the retailer.
On the other hand, if Ki > K̄i , then the firm does not find it profitable to offer the APD and the
usual wholesale price contract ensues. The following theorem formalizes this equilibrium outcome.
Theorem 2. Equilibrium Outcome
(1) Actions: The tuple of actions {δi∗ , Q∗i , Pi∗ } characterizes a perfect Bayesian equilibria of
the setup just described. For all i,
0
0
0
0
δi∗ = δ i , Q∗i = µi + σi zδi , Pi∗ = µi + σi zγ
if Ki ≤ K̄i ,
δi∗ = 0,
otherwise.
Q∗i = 0,
Pi∗ = µi + σi zγ
(2) Profits: The equilibrium expected profits for retailer i are EΠR
i , and the equilibrium expected
P
F
profits for the firm, EΠF are N
i=1 EΠi , where
0
EΠR
i = (Ri − wi ) µi ,
EΠFi = (wi − γCL ) µi − CL φ (zγ ) σi − Ki
if Ki ≤ K̄i ,
EΠR
i = (Ri − wi ) µi ,
EΠFi = (wi − γCL ) µi − CL φ (zγ ) σi
otherwise,
and where zγ , zδ , δ i , and K̄i are as defined in the previous lemmas.
The setup of this model effectively includes a take-it-or-leave-it APD offer, which gives the firm firstmover advantage and all the bargaining power. Consequently, in equilibrium, the retailer obtains
the same expected profits as not participating in the APD scheme while the firm appropriates all
the supply chain benefits, as demonstrated in Theorem 1. Yet Theorem 1 also suggests that, with
alternate bargaining and negotiation mechanisms, the equilibrium contract may be more egalitarian
in that it could strictly improve every agent’s expected profits.
In Section 4 we will develop a more realistic model of this setup, where the firm will still have the
same first-mover advantage but would no longer have full knowledge about each retailer’s operations.
We will see that such a setup also leads to some restoration of balance; indeed, both the retailers
and the firm can have strictly positive benefits from participating in the APD scheme.
In the model just described, the firm has an additional advantage that may not extend to other
situations: because the products are customized for each retailer/market, the firm can set retailerand market-specific discounts. Our analysis in Section 7 examines such a setup, again with reduced
discrimination power for the firm. That is, the firm can offer only one discount to all retailers, so
the retailers derive strictly positive benefits from the supply chain.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
21
Our model can also be simply modified to capture the case where the administrative cost actually
accrues to the retailer. Under this interpretation, the cost Ki becomes the retailer’s required participation incentive. Corbett et al. [2004] consider such a participation incentive in their model.
The qualitative results described previously will continue to hold, and retailers will derive positive
benefits from participating in the APD.
4. Retailers’ Costs/Information Precisions are Unobservable
In the model described so far, the firm’s decision to offer the advance purchase discount scheme
(Theorem 2) and the extent of the discount to be offered (Theorem 1) both depend crucially on the
retailer’s administration cost Ki and the quality of the retailer’s private information ti , which trans0
lates into the posterior standard deviation of demand, σi . The administrative cost and the quality
of the retailer’s private information jointly influence the retailer’s minimum acceptable discount
(MAD) δ i . In the previously described equilibrium, the firm offers this minimum acceptable level to
the retailer, and consequently all the benefits of the APD scheme end up accruing to the firm. Our
assumption in the preceding analysis was that these two parameters are common knowledge—in
other words, the firm was fully aware of the retailer’s operations and thus knew both the administrative cost and the quality of the private information that the retailer could provide. In practice,
these variables are rarely known with certainty by upstream firms. Administrative cost depends on
the retailer’s internal mechanisms, organizational inertia, and other costs that are incurred at the
retailer end. Thus, it is unreasonable to suppose that the upstream firm knows them with certainty.
Similarly, the quality of the retailer’s information is inherently a function of the kind of market
research conducted by the retailer and the efficacy of her information systems, and as such it is not
visible to upstream firms in the normal course of business (Taylor and Xiao 2009).
In this section we extend the analysis of previous sections to a more realistic setup, where the firm
does not accurately know the retailer’s minimum acceptable discount (MAD). This setup presents us
with a few key questions: (i) How does the firm implement an efficient advance purchase discount
scheme even while being handicapped by the lack of knowledge about the retailers? (ii) Do the
benefits of Advance Purchase Discounts that are derived from information sharing persist in such a
setup?
Note that unobservability of both the information quality and the administrative cost translate
equivalently into an uncertainty about the minimum acceptable discount (Equation 3.2). In the
22
WENJIE TANG AND KARAN GIROTRA
interest of brevity, in this section the analysis assumes that the firm does not know the administrative
cost and that alone leads to the uncertainty about the MAD. The intuition follows similar lines if
it were the uncertain information quality that alone drove the uncertainty about the MAD.
4.1. Retailers’ and Firm’s Strategies. The retailers’ profit function and strategy profile do not
change from the setup described in Section 3. The firm, however, faces a new challenge in computing
the discount to offer. Essentially, it faces a trade-off between the likely results of a deeper discount
and a more modest one: a modest discount would yield the firm higher benefits from the APD
scheme, though it would reduce the likelihood of retailer participation; whereas a deep discount
would increase the chances of retailer participation but at the cost of reduced benefits to the firm.
The following example describes the essence of this trade-off and the corresponding strategy for the
firm in a simple setting where the retailer’s administrative cost could be one of two possible values.
Example: Retailer’s administrative cost takes one of two values. Consider a setting where
the unobservable administrative cost of retailer i can take one of two values: Ki1 with probability
b or Ki2 with probability 1 − b, where 0 ≤ Ki2 ≤ Ki1 ≤ K̄i .6 By Equation (3.2), retailer i’s
corresponding MAD becomes either δ i1 or δ i2 , where 0 ≤ δ i2 ≤ δ i1 ≤ 1. The following lemma
describes the firm’s strategy.
Lemma. The firm chooses the discount level to be δ i1 if b K̄i − Ki1 ≥ (1 − b) (Ki1 − Ki2 ) and δ i2
0
otherwise. Recall that, as before, K̄i = CL φ(zγ )(σi − σi ).
Setting a discount δi is thus a trade-off between the potential loss of benefits due to an “excessive”
discount and the risk of losing the retailer’s participation and thereby all the APD benefits. If the
firm sets the discount at δ i1 then there is no risk of losing the retailer’s participation and the APD
benefits, but there is a loss of benefits due to an excessive discount. This arises when the firm sets
the discount at δ i1 when δ i2 would have been sufficient (when the retailer’s true cost is Ki2 ). The
extent of this loss is Ki1 − Ki2 , and the probability of this contingency is 1 − b. This consequence
of setting the discount at δ̄i1 is captured by the RHS of the critical inequality in the lemma just
stated.
6If K > K̄ and K ≤ K̄ , then the firm’s optimal strategy would be to simply set the discount at δ ; and if
i1
i
i2
i
i2
Ki1 ≥ Ki2 > K̄i , the firm does not offer the APD. Here, we display results for the most interesting non-trivial case
where 0 ≤ Ki2 ≤ Ki1 ≤ K̄i .
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
23
On the other hand, if the firm sets the discount at δ̄i2 , then it never offers an excessive discount
but may lose its share of the benefits from the APD scheme. This happens when the true required
discount is δ̄i1 (i.e., when Ki = Ki1 ), which happens with probability b. In such a case, the firm
loses its share of the benefits from the APD; as before, these benefits are equal to K̄i − Ki1 . The
total expected consequence of this loss is captured by the LHS of the critical inequality in our
lemma. The appropriate discount is determined by comparing the expected excessive discount loss
from setting the discount at δ̄i1 and the expected loss of APD benefits from setting the discount at
δ̄i2 .
We now extend the intuition from this example to a case where retailer i’s administrative cost Ki
follows a distribution with a general pdf gi (·) and cdf Gi (·), and positive support in the interval
[Ai , Bi ]. Further more, we assume that Gi (·) has a decreasing reversed hazard rate (DRHR).7 The
firm’s strategy is captured by the following theorem.
Theorem 3. Firm’s Strategy Profile: The firm offers each retailer a discount δˆi :
δ̂i = δ̂i (k̂i )
if k̂i ≤ K̄i ,
δ̂i = 0
otherwise.
Here
n o
0
δˆi (k̂i ) = δi δi wi µi − wi φ (zδi ) σi = k̂i ,
(4.1)
k̂i = 1Ai ≤ki0 ≤Bi ki0 + 1ki0 <Ai Ai + 1ki0 >Bi Bi ,
and
(4.2)
)
( K̄ − k g (k )
i
i
i
i
=1 .
ki0 = ki Gi (ki )
Theorem 3 illustrates that, when it cannot observe the retailer’s administrative cost, the firm still
treats the situation as if the retailer’s administrative cost is known and takes a certain value.
Essentially, the firm uses a “certainty equivalent” value of the unobservable administrative cost and
7DRHR is equivalent to log-concavity for a cdf (Chandra and Roy 2001); the assumption is not at all restrictive
because it includes most common distributions. Distributions with increasing failure rate—such as the Weibull,
gamma, normal, uniform, and log-normal distribution—are all found to be DRHR distributions (Block et al. 1998).
24
WENJIE TANG AND KARAN GIROTRA
then proceeds to set the discount in exactly the same fashion as in Section 3.8 The optimal value
for this “certainty equivalent” is defined in Equation (4.1) and (4.2). Not surprisingly, it is closely
related to the firm’s beliefs about the distribution of Ki ’s value (i.e., gi (·), Gi (·), and the support
[Ai , Bi ]). Moreover, as discussed in Section 3, if the optimal “certainty equivalent” is no more than
the maximum benefits the firm could receive from employing the APD scheme, K̄i , then the firm
offers the APD to the retailer with the corresponding optimal discount δ̂i (k̂i ); otherwise, the firm
does not offer the APD.
The foregoing analysis characterizes the firm’s optimal strategy when the administrative cost is
unobservable. Next we compare this optimal strategy with the strategy computed in the preceding
section, where the administrative cost was common knowledge.
Recall that, when Ki is observable to the firm, the optimal discount is influenced only by the
magnitude of Ki and that the higher Ki , the higher the optimal discount. This is no longer true
when the firm cannot observe Ki , as illustrated by the following example.
Example: Effect of the Degree of Uncertainty about Retailer’s Cost. Consider three
different retailers i = 1, 2, and 3, where the administrative cost Ki is either li or hi with equal
probability. According to Theorem 3, the optimal discounts to offer are as computed in Table 1.
Retailer li hi δ̂i ×10−3
1
0.1 0.2
11
2
0.1 0.3
16
3
0.1 2
5
Table 1. Discounts for three retailers with different administrative cost distributions. The results
0
are calculated assuming CL = 1.8, wi = 1.9, γ = 0.5, µi = 10, σi = 3, and σi = 0.1.
The administrative costs for both retailer 2 and retailer 3 are, on average, higher than that of retailer
1, and they are also larger in a first-order stochastic dominance sense. The logic of Section 3 would
suggest that the discount offered to either retailer 2 or 3 would be higher than the discount offered
to retailer 1. This logic holds as far as retailer 2 is concerned; however, the optimal discount offered
to retailer 3 is actually smaller than that offered to either retailer 1 or 2! This example illustrates
that neither the mean of the unknown administrative cost nor first-order stochastic dominance
fully determines the order of the optimal discount. The uncertainty about the estimate of the
8A conceptually similar treatment of a different context is presented in Tang et al. [2009].
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
25
unobservable administrative cost is the missing explanatory variable. We next examine its impact
on the firm’s optimal strategy.
Write retailer i’s administrative cost as Ki = ui + si i with support [Ai , Bi ], where i is a random
variable with DRHR, standard deviation one, and cdf Gi0 (i ) over support [(Ai −ui )/si , (Bi −ui )/si ].
Then the distribution of Ki has cdf Gi (ki |ui , si ) = Gi0 ((ki −ui )/si ) and a decreasing reversed hazard
rate. The standard deviation of the distribution Gi (ki |ui , si ) equals si , and increasing (decreasing) si
corresponds to stretching (contracting) around ui . In that sense, si serves as a measure of the firm’s
uncertainty about retailer i’s administrative cost Ki . Observe that when si = 0, the distribution
of Ki is degenerate with the mass point at ui , and then the optimal discount to offer equals δ̂i (ui ).
The next theorem shows that if si is small, the optimal discount is higher than δ̂i (ui ), and if si is
large, the optimal discount is lower than δ̂(ui ).
Theorem 4. Denote by k̂i (ui , si ) the solution to Equation (4.2) for a distribution with cdf Gi (ki |ui , si ) =
Gi0 ((ki − ui )/si )). The optimal discount is δ̂i (k̂i (ui , si )). If si ≤ si0 then δ̂i (k̂i (ui , si )) ≥ δ̂i (ui ), and
if si > si0 then δ̂i (k̂i (ui , si )) < δ̂i (ui ). Here si0 = (K̄i − ui )gi0 (0)/Gi0 (0).
Theorem 4 points out that, if the uncertainty about the retailer’s administrative cost is small then
the optimal discount is higher than in the case of no uncertainty; conversely, if the uncertainty is
large then the firm should set a lower discount than in the case of no uncertainty.
This discussion offers the firm an important practical guideline for setting the discount when it faces
a retailer about whom it has limited knowledge. The key to setting the appropriate discount lies not
only in the estimate of the unobservable administrative cost but also in the uncertainty around it.
In essence, when the uncertainty about the retailer’s administrative cost is small, a higher discount
should be given than in the case of no uncertainty; when the uncertainty is large, a lower discount
is optimal. In our experience with implementing APD schemes at real firms, we find that very often
the firm does not know the retailer’s administrative cost but does have some sense of its typical
level and of the uncertainty around this estimate. Armed merely with these two intuitive measures
and the insights from our foregoing discussion, the firm has a strong guideline on how to proceed in
offering these discounts within its supply chain. We shall illustrate one such experience of applying
APDs in Section 5. Next, we examine equilibrium outcomes under limited knowledge.
4.2. Equilibrium Outcome. The equilibrium outcome follows from the strategy profiles just described for the retailers and the firm. If k̂i ≤ K̄i then the firm finds it profitable to offer an Advance
26
WENJIE TANG AND KARAN GIROTRA
Purchase Discount, and if k̂i ≥ Ki then retailer i finds it in her interest to purchase some quantities
in advance. This purchase signals the retailer’s private information. If k̂i > K̄i or k̂i < Ki then
either the firm does not find it profitable to offer the APD scheme or retailer i won’t participate
given an insufficient discount. As a result, the usual wholesale price contract outcome ensues. The
following theorem formalizes this equilibrium outcome.
Theorem 5. Equilibrium Outcome
n
o
∗
∗
∗
(1) Actions: The tuple of actions δ̂i , Q̂i , P̂i characterizes a perfect Bayesian equilibria of
the setup just described. For all i,
δ̂i∗ = δ̂i , Q̂∗i = 0,
0
P̂i∗ = µi + σi zγ
0
0
0
if k̂i ≤ K̄i and k̂i < Ki ,
δ̂i∗ = δ̂i , Q̂∗i = µi + σi zδi , P̂i∗ = µi + σi zγ
if k̂i ≤ K̄i and k̂i ≥ Ki ,
δ̂i∗ = 0,
if k̂i > K̄i .
Q̂∗i = 0,
P̂i∗ = µi + σi zγ
(2) Profits: The equilibrium expected profits for retailer i are EΠ̂R
i , and the equilibrium expected
P
F
profits for the firm EΠ̂F are N
i=1 EΠ̂i , where
0
EΠ̂R
i = (Ri − wi ) µi + k̂i − Ki ,
EΠ̂Fi = (wi − γCL ) µi − CL φ (zγ ) σi − k̂i
if Ki ≤ k̂i ≤ K̄i ,
EΠ̂R
i = (Ri − wi ) µi ,
EΠ̂Fi = (wi − γCL ) µi − CL φ (zγ ) σi
otherwise,
where k̂i and δ̂i are as defined in Theorem 3.
When the retailer’s administrative cost is unobservable, the firm treats the retailer as if her true cost
is k̂i . The firm then offers the equilibrium discount in the same fashion as when Ki is observable,
and all the resulting expected profits and actions retain the same forms as in Theorem 2 but with
Ki replaced by its “certainty equivalent” k̂i . However, two differences should be noted here. First,
since k̂i might be lower than the retailer’s true administrative cost, the retailer’s participation is
not ensured even when the firm finds it profitable to incentivize the retailer’s participation with an
appropriate discount δ̂i∗ . This means that each retailer’s participation condition must be considered
even when a sufficiently deep discount is offered, which leads to the first inequality of the new
condition Ki ≤ k̂i ≤ K̄i . Similarly, there is also a chance that k̂i is greater than the retailer’s true
administrative cost, in which case the retailer could earn a positive benefit under equilibrium; this
leads to the term k̂i − Ki in the retailer’s new expected profits.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
27
4.3. Benefits of Advance Purchase Discounts under Unobservability. In this section, we
investigate how the firm’s limited knowledge about retailer parameters influences the benefits arising
from APDs. We describe the dynamics formally in the next theorem.
Theorem 6. Equilibrium Benefits: If the retailer’s administrative cost Ki is unobservable to the
firm then, on the equilibrium path, the participating retailers, the firm, and the supply chain are all
better off under APDs. The benefits are given as
∆EΠ̂R
=
1
k̂
−
K
i
i ≥ 0,
i
Ki ≤k̂i ≤K̄i
F
∆EΠ̂ =
N
X
1Ki ≤k̂i ≤K̄i K̄i − k̂i ≥ 0,
i=1
S
∆EΠ̂ =
N
X
1Ki ≤k̂i ≤K̄i K̄i − Ki ≥ 0,
i=1
where k̂i is defined as in Theorem 3.
Theorem 6 shows that, when the APD scheme is implemented, every participating agent in the
supply chain is better off. This result differs from the case where retailers’ administrative costs are
observable to the firm in two important ways. First, when administrative cost is unobservable to the
firm, the retailer has some further private information that allows her to extract some information
rents k̂i − Ki from the firm, which consist of the difference between her “certainty equivalent”
administrative cost and her real one. As a result, instead of setting the discount at each retailer’s
MAD to appropriate all the benefits accruing to the supply chain from the information sharing
K̄i − Ki , the firm now gets K̄i − k̂i and the remaining benefits k̂i − Ki go to the retailer. Second,
given unobservability, even when the firm wants the retailer to participate in the APD scheme and
sets the appropriate discount δ̂i∗ , it remains possible that this discount will turn out to be insufficient
from the retailer’s perspective. In such cases, the retailer does not participate and the supply chain
does not appropriate the potential benefits from the APD. The indicator 1Ki ≤k̂i ≤K̄i takes this into
account. The incidence of such outcomes reduces the expected gains to the supply chain from
employing the APD scheme, though they still remain positive. In short, the unobservability leads
to a more egalitarian distribution of the gains from the APD scheme, but the total magnitude of
the gains is somewhat reduced.
28
WENJIE TANG AND KARAN GIROTRA
Note here that, under unobservability, the firm’s incentive to set the discount is not aligned with the
supply chain’s incentives. The supply chain prefers the highest possible discount so as to include
all the retailers with nonnegative K̄i − Ki . Yet the firm has an additional goal in setting the
discount: it cares not only about ensuring the participation of all information-rich retailers and
maximizing the total surplus generated by the APD contract but also about the share of supply
chain benefits that it can appropriate. This second concern is driven by the uncertainty about the
retailers’ administrative costs. The following corollary illustrates the effect of this uncertainty.
Corollary. Under the implementation of the APD scheme with discount δ̂i∗ characterized in Theorem
4 and 5:
(i) if si ≤ si0 , then ∆EΠ̂R k̂i (ui , si ) ≥ ∆EΠ̂R (ui ) and ∆EΠ̂F k̂i (ui , si ) ≤ ∆EΠ̂F (ui );
(ii) if si > si0 , then ∆EΠ̂R k̂i (ui , si ) < ∆EΠ̂R (ui ) and ∆EΠ̂F k̂i (ui , si ) > ∆EΠ̂F (ui ).
Recall that ∆EΠ̂R k̂i (ui , si ) and ∆EΠ̂F k̂i (ui , si ) represent the retailer’s and firm’s respective
benefits from the APD scheme under unobservability, and ∆EΠ̂R (ui ) and ∆EΠ̂F (ui ) represent
those benefits under full observability. The corollary shows that, when the firm’s uncertainty about
the retailer’s administrative cost is small, its benefits from the implementation of an APD scheme are
lower than if it knew her costs with certainty. At the same time, the retailer’s benefits increase when
the uncertainty about her administrative cost is small. However, when the uncertainty becomes
larger than a threshold value, the retailer’s benefits from the APD scheme are lower than in the full
observability case while the firm’s benefits become higher. At this point, the firm is too uncertain
about the retailer and essentially loses interest in attempting to induce her participation.
This result offers two insights into the strategic behavior of both the firm and the retailer. From
the firm’s point of view, it is beneficial to collaborate with a retailer about whom it has either full
knowledge or no knowledge at all. In either case, the optimal discount offered would be a very
modest one and thus would not hurt the firm much in benefits. This all-or-nothing policy shares
the same intuition—though in a different setting—regarding how a manufacturer should choose its
forecasting newsvendor described in Taylor and Xiao [2008].
On the retailer’s side, the corollary provides the retailer an incentive to strategically communicate
information about her administrative cost. The retailer will receive the deepest discount when
the firm’s uncertainty about her administrative cost is neither too big nor too small. In order to
encourage the firm to set a high discount, the retailer should not completely hide her administrative
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
29
cost information from the firm, since doing so would lead the firm to react with a safe strategy:
setting a modest discount to ensure that it gains a large part of the APD benefits. Nor should
the retailer communicate her cost information fully, since then the firm would take full advantage
and set the discount as if the retailer’s MAD were observable. A conceptually similar dilemma
is observed in a different context by Kim and Netessine [2009]. In their paper, a manufacturer
and a supplier collaborate in the product design stage to reduce the uncertainty about component
production cost; the supplier fears revealing his proprietary cost information and thus is reluctant
to fully collaborate.
5. Implementation at Costume Gallery
In this section, we report on the implementation of our proposed APD scheme at a New Jersey–
based fashion apparel wholesaler that sources extensively from Asia and retails through an expanding
U.S. network. This allows us to (i) validate the assumptions and setup of our model, (ii) illustrate
an implementation of APDs, and (iii) enumerate the benefits of applying an APD scheme. The
interested reader is referred to Girotra and Tang [2009] and Girotra and Tang [2010] for a detailed
description of Costume Gallery’s business context and the benefits from implementation of APDs.
5.1. Background of Costume Gallery. Costume Gallery is a privately held, New Jersey–based
wholesaler of dance costumes. In the U.S. market, it is among the top three wholesalers of dance
costumes; sales in 2005 amounted to about US$30 million.9 Costume Gallery has been family
run since its inception in 1957; the third generation of the family took over in 1997 and has been
instrumental in bringing scientific management principles to the enterprise.
Costume Gallery was founded on the premise of excellent customer service. The strategic focus at
Costume Gallery is on fully satisfying its customers at all costs. This strategic focus translates into
a wide choice of customizable costumes and a target 100% fill rate. At any time, Costume Gallery
has as many as 500 styles in its catalog; if a requested style is not in stock, then Costume Gallery
can produce it to spec for the customer.
Costume Gallery’s supply chain is illustrated in Figure 4. Costume gallery sells most of its merchandise through dance schools. Typically, the end customer is a student enrolled in dance classes
at dance schools. The instructor at a dance school plans a dance production and then decides on
9Costume Gallery is privately held and does not disclose its financial performance. This number is the authors’
estimate.
30
WENJIE TANG AND KARAN GIROTRA
Figure 4. The Dance Costume Business
the appropriate costumes. The school then procures the costumes from Costume Gallery and sells
them to the students at a small markup. Despite this markup, dance schools do not usually hold
any inventory or bear any of the supply chain risks; all inventory costs and risks are traditionally
borne by Costume Gallery.
Costume Gallery can produce the dance costumes in-house and can source their production (domestically or overseas). Costumes produced in-house cost about $14 on average and can be produced
with a lead time of 1–2 days. Very small lots can be produced at Costume Gallery. As an alternative, costumes can be sourced—usually from Asia. Such fashion apparel is typically sourced at
a total cost that is just 30%–40% of the western production cost; however, the offshore production
and shipping lead time can be 2–3 months. Because there is a substantial cost saving in sourcing
from Asia, Costume Gallery would prefer to source as much of its production from Asia as possible.
Costume Gallery operates two business units: one deals in generic dance products and the other
procures products that are customized to meet dance school requirements. Both generic and customized products can be manufactured at the in-house or Asian facilities. Because of product design
constraints and the difficulty of finding skilled labor, Costume Gallery does not use any form of
postponement for the customized products.
The dance costume business is highly seasonal: 90% of the annual demand occurs in the second half
of April, which coincides with the end-of-school-year dance performances. The timeline of the dance
costume business is illustrated in Figure 5. Costume Gallery typically starts designing costumes in
July of the previous year. These costumes are profiled in a catalog that is sent out to dance schools
in early August. Dance schools finalize their enrollment in November and, by December, they have
a good idea of the theme for the dance performance in April. Therefore, the schools have potential
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
31
Figure 5. Timeline of the Dance Costume Business
access to information on the demand for different dance costumes: they can restrict the size of their
classes and fix the theme of the dance. This may come at the cost of additional business, but if
the schools wanted to, they can do this. However, there is still some residual uncertainty caused
by changing sizes of students and by additions to or dropouts from classes. Given this residual
uncertainty, schools behave strategically and do not place any orders with Costume Gallery at this
time.
To meet the April demand, Costume Gallery must place its overseas orders by January in view
of the 2–3-month lead time required. Typically, these orders must be placed before the Chinese
New Year holiday, which occurs in late January or early February. These orders are placed on
the basis of the best information that Costume Gallery has about the possible demand from each
dance school. Dance schools have a far better idea of the likely demand but, because of their
residual uncertainty about this demand, they have no incentive to place any orders early in the
season. Therefore, Costume Gallery’s overseas orders are placed without using the demand-relevant
information available to dance schools. From January to April, Costume Gallery obtains increasingly
accurate demand information and uses its in-house production (and local contractors) to meet
demand above the quantities already sourced from Asia.
5.2. Application of APDs at Costume Gallery. The supply chain and business cycle of Costume Gallery display the essence of our stylized model. Costume Gallery corresponds to the firm
in our model and the dance schools to its network of retailers. Like the firm, Costume Gallery has
two production modes: one cheaper with longer lead time (Asia); the other more expensive with
shorter lead time (in-house). Like the retailers, dance schools have access to superior knowledge
about future demand. In the absence of advance purchase discounts, however, the schools have
32
WENJIE TANG AND KARAN GIROTRA
no incentive to either gather this superior knowledge or place an early order; thus the information
is never shared and utilized to improve supply chain efficiency. Our theoretical model suggests
that, by implementing Advance Purchase Discounts, Costume Gallery can elicit accurate, timely
information from the dance schools that can then be used to improve sourcing decisions.
5.3. Estimation of the Benefits from APDs. In this section, we provide a rough estimate of
the potential benefits that Costume Gallery can obtain if the advance purchase discount scheme
suggested by our theory is implemented. To estimate this number, we used actual data from the
operations of Costume Gallery to the greatest extent possible. We have made every effort for these
numbers to be representative and applicable to a wide set of apparel businesses. Nonetheless, as
with any case-based data, the numbers estimated in this section should only be viewed as indicative
of the potential gains possible; they should not be construed as a rigorous empirical estimate for a
typical firm in this industry.
For ease of illustration, we restrict our attention to the customized products business unit at Costume Gallery. Typically, Costume Gallery does not have extensive knowledge about the operations
of its retailers, the dance schools. In particular, it is difficult for Costume Gallery to assess the exact
administrative order costs that each dance school would incur in placing an early order. Thus, the
theoretical results from Section 3 on customized products, and from Section 4 on limited information
are most applicable to our setup.
In order to estimate the benefits from our proposed strategy, we consider a subset of costumes and
dance schools. There is almost no difference in the operational economics of different costumes in
this subset (or between the vast majority of costumes). Hence we build this rough estimate using the
economics of a representative costume, “Dream Ballet”. The design details, required workmanship,
level of customization, and costs of materials for this costume are all fairly representative of a typical
product at this firm.
Each Dream Ballet costume is offered to a dance school at a price of $35 (wi in our model), whereafter
the dance school normally charges a $5 commission and sells it to each student at a price of $40
(Ri ). It costs Costume Gallery $14 (CL ) to produce such a costume in-house; however, if Costume
Gallery outsources the production to Asia, then the total cost—including shipping and all other
incidentals—is estimated to be $7 (γ = 0.5).
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
33
The demand Di from our model corresponds to the number of units demanded, by a particular dance
school, of a particular costume (style) in a particular size. This corresponds to demand for one SKU
to be sourced. We refer to this as an order, and we refer to Di as the “order size”. To estimate a
forecast distribution for Di , we employ the A/F ratio method discussed in Cachon and Terwiesch
[2006]. We estimate Costume Gallery’s order demand forecast to be normally distributed, with a
mean of 44.3, and a standard deviation of 19.4 units. Next we must obtain the demand forecasts that
the dance schools construct, which are needed to estimate the posterior distribution of demand that
Costume Gallery could construct if they obtained early order information. This, in turn, requires the
forecasts that schools would construct if they could avail themselves of Advance Purchase Discounts.
In absence of APDs, the schools do not construct any forecasts and so the data needed to estimate
this distribution do not exist. However, based on the authors’ conversations with the management
at Costume Gallery, we estimate that the forecast errors can probably be reduced by about 50% if
it used advanced information on each school’s enrollment size and dance theme (both of which are
school-level private information). Hence, we assume that the posterior demand distribution has a
0
standard deviation that is half that of the prior demand distribution—in other words, σi = 10.
In order to implement the proposed APD scheme, Costume Gallery must calculate the administrative
costs for dance schools to participate. The administrative costs depend on dance schools’ internal
mechanisms, organizational inertia, and other costs incurred at their end to acquire the information.
These include lost revenues due to restricting the class size, fixing the dance theme, the costs of
capital required, labor costs for measuring students an additional time (students sizes often change
during the school year). Costume Gallery does not have a good idea of these costs. Moreover, they
believe that these costs differ widely across different schools.
Some schools are located in low-cost regions and are staffed by cheaper teachers. Other schools
employ professional dancers as teachers, and their costs are expected to be higher. In addition,
Costume Gallery has a better sense of the costs at schools with which it’s had a long collaboration
than at schools that are new clients. To deal with this heterogeneity in school types, we consider
three different classes of schools. Type A are high cost schools with which Costume Gallery has
a long-standing relationship, type B are high-cost schools that have recently begun purchasing
from Costume Gallery, and type C schools are low-cost schools. Costume Gallery estimates the
administrative cost per order for each of the three school types separately, as summarized in Table
2.
34
WENJIE TANG AND KARAN GIROTRA
School Type
A
B
C
Characteristics
Estimate of Order
Costs ui
Standard Deviation of
Estimate of Order
Costs si
High Cost, Long Relationship
40
High Cost, Short Relationship
40
Low Cost
20
Table 2. Dance Schools’ Administrative Costs
4
25
25
School Type
# of Orders from
Schools in This
Category
Discount
Percentage of Schools
that Avail APDs
Expected Increase in
Profits for Costume
Gallery (US$)
A
B
C
25
29
102
5.14%
3.74%
2.76%
75.78%
33.65%
50.93%
183.84
224.95
1658.09
Total Increase in Profits (US$)
Pre-APD Net Profits (US $)
Increase in Net Profits (%)
2,066.88
12, 093.90†
17.09%
Table 4. Estimated Benefits of Advance Purchase Discounts
†Profits on sales of US$0.24 million covered by this illustration (computed assuming 5% net profitability of the firm).
Using the results from Theorems 3 and 6, we estimate the benefits for Costume Gallery of employing
Advance Purchase Discounts; the results are reported in Table 4. Our estimates indicate that
Costume Gallery could increase its net profits by about 17% if it utilized APDs. Although we
provide this analysis for only a subset of schools and products, we believe that these numbers are
representative of Costume Gallery’s product lines and also of other firms in similar industries.
6. Conclusion
This study proposes the mechanism of an Advance Purchase Discount contract that enhances the
effectiveness of a firm’s global sourcing strategy by incentivizing accurate information communication from downstream tiers in its supply chain at a time chosen by the firm. The strategy involves
offering a discount to downstream tiers for orders that they place before a specific time, which is
usually chosen by the firm to coincide with the last possible chance for overseas orders. Acting in
their own best interests, the downstream retailers use the best information accessible to them in
making the early orders. We show that observing these orders enables the upstream firm to infer
the information available to the downstream retailers; then the firm can make its global sourcing
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
35
decisions using the demand information inferred. The principal effect of the early order opportunity is to synchronize the timeline of orders placed by retailers with the timeline of decisions that
the global sourcing firm must make. This alignment drives the timely communication of accurate
demand information from the downstream retailers to the firm. We demonstrate the benefits of
this scheme, identify the conditions under which it is most potent, and isolate the benefits arising
from improved information sharing and redistribution of risk. We then demonstrate that, taken
together, these benefits can improve the performance of each agent in the supply chain and also of
the supply chain as a whole. We demonstrate the robustness of these results by considering two
different product designs: one where products are customized for each market and one where the
same generic product is sold in all markets.
We also examine the application of these contracts in a realistic setting. We analyze our proposal for employing APD contracts in the context of practical constraints, as when a firm has
limited knowledge about its retailers’ operations. We characterize the optimal strategy in terms of
a “certainty-equivalent” value of the unobserved parameter, a value that the firm can use as if it
knew the unobserved parameter for sure. We remark that this value leads to an asymmetric and
“degree-of-unobservability dependent” departure from the full-observability design of the Advance
Purchase Discount. When the uncertainty in the unobserved parameter is small, the optimal discount is higher than in the case of full observability; and when the uncertainty is large, the firm
should set a lower discount. It is significant that this analysis affirms that the benefits of APDs
(improving each agent’s profits as well as the supply chain’s profits as a whole) persist even under
this practical constraint.
To validate the applicability of our analytical setup, estimate the benefits accruing from our prescriptions, and develop practical guidelines for application, we close this study by reporting on the
implementation of our APD scheme at Costume Gallery, a small-scale U.S.-based firm in the apparel business. We find that our model accurately captures the business context for this firm. The
proposed contract shares its structure with commonly used payment terms and thus found limited
resistance in implementation. Further, using data derived from the operations at Costume Gallery,
we estimate that applying the APD scheme could yield a 17% increase in the firm’s net profits.
Our proposal for APD contracts suffers from several key limitations. Designing an optimal APD
contract requires the estimation of parameters related to retailers’ operations to an infeasible degree
of accuracy. Our analysis of the case with limited knowledge of retailer operations addresses these
36
WENJIE TANG AND KARAN GIROTRA
concerns to a certain extent; however, for a productive implementation of an APD scheme, the firm
must have extensive archival data on its own and its retailers’ forecasting abilities, as well as good
cost accounting systems, in order to measure accurately the benefit of sourcing overseas. This may
not be possible for many firms. Further more, computing and administering these discounts require
managerial skills that may not be available to many small organizations. Finally, the benefits of this
scheme are highly sensitive to the extent of the discount offered. Yet practical constraints arising
from industry practice and long-held relationships with partners prevent the firm from implementing
an optimal version of the APD contract, and this may significantly reduce the accrued benefits.
This study provides several new directions for future research. Our preliminary analysis assumed
that the demand for multiple products and from multiple retailers is uncorrelated. However, in
practice we find demand to be correlated significantly across different products and across different
markets. In such a situation, early order information would be useful for improving the demand
forecasts of multiple products and markets. In the absence of modeling such dynamics, our current
analysis underestimates the value of information acquired from Advance Purchase Discounts and
consequently underestimates their benefits. A study taking into account the correlation structure
of demand and the consequent dynamics holds significant promise.
Our current Bayesian information structure with conjugate demand-signal pairs, while theoretically
appealing and analytically tractable, does not capture an important facet of information acquisition:
the dependence of the quality of information obtained on the extent of the discount. An alternate
modeling setup that captures this real-life feature is likely to provide useful insights into the science
of administrating APDs.
Another avenue for future research is in estimating parameters that allow for effective administration
of Advance Purchase Discounts. As we have emphasized, knowledge of many retailer side parameters
is critical in this analysis. As in our illustration of the Costume Gallery, some parameters (such as
the retailers’ forecasting abilities) are hard to observe directly even with extensive archival databases.
Perhaps a sophisticated empirical strategy can be found to impute some of these parameters from
other demand-related actions by the retailers. Such a strategy holds significant promise and could
make widespread implementation of APDs a reality.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
37
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Appendix
7. Benefits of Advance Purchase Discounts: Generic Products
Here we present an alternate version of the model of Section 3. In this model, each retailer sells an identical good
and so, unlike in Section 3, the goods are not customized for each retailer/market. This introduces several changes
in the model setup. First, since the product sold to each retailer is identical, the wholesale price for each retailer
must be the same; we denote this price by w. And instead of tailoring a different discount δi for each retailer, the
firm must offer the same discount δ to all retailers. Second, since the same product is sold through all retailers, the
firm can procure and inventory the same product for all retailers. Essentially, instead of ordering Pi for each retailer,
the firm places an overseas order of P units for all retailers. Third, we also relax the assumption of Sections 3 and
4 that the retailers cannot return or cancel the advance orders. We now allow a return policy: after demand Di is
realized, retailer i can either place additional orders (Di − Qi )+ at w or cancel (Qi − Di )+ from purchased quantity
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
41
and get back v for each canceled unit, where 0 < v < δw to ensure that the retailers would still be penalized for
over-ordering. To focus on the benefits of Advance Purchase Discounts, we assume here that v is exogenously given
such that the firm’s expected profits are greater under a return policy than otherwise.10
As in previous sections, we first provide the retailers’ and firm’s strategy, followed by the equilibrium outcome and
the benefits of the APD scheme.
Lemma. Retailers’ Strategy Profile: Each retailer orders an advance purchase quantity Qi and earns expected
profits EπiR , where
0
0
EπiR = (Ri − (1 − δ) w) µi − (w − v) φ (zδ ) σi − Ki
if δ ≥ δ i ,
Qi = 0,
EπiR = (Ri − w) µi
otherwise,
n o
0
δ i = δ δwµi − (w − v) φ (zδ ) σi − Ki = 0 ,
(7.1)
and zδ = Φ−1
0
Qi = µi + σi zδ ,
wδ
w−v
.
Regardless of the product characteristics, retailers are still independent decision makers and so their strategy profile
remains the same as in Section 3. The only difference stems from our introduction of the return policy: because
retailers can now cancel their orders and recover v for each canceled unit, essentially they share less risk than in the
0
customized product case. This reduced risk share is captured by (w − v) φ(zδ )σi .
At the same time, several changes occur at the firm’s end. First, owing to the generic nature of the product, the same
wholesale price and discount are set for all retailers; moreover, the firm can procure and inventory the same products
for all retailers, which has a risk pooling effect on demand uncertainty. Second, the introduction of a return policy
further enlarges this risk pooling effect. In particular, after demand is realized, instead of re-ordering the shortfall
P
+
N
(Di − Pi )+ for each retailer i, the firm now can redirect the returned units and so need only re-order
i=1 Di − P
units. Finally, recall that the firm in our original model could offer a tailored discount δi to each retailer and so the
decision to offer the APD scheme depends solely on the characteristics of the retailer in question. However, when
the firm must use a single level of discount δ, this decision is influenced by the entire population of retailers. In
particular, for a given discount δ, all retailers whose MADs (from Equation 7.1) are less than δ will participate in
this APD scheme.
The choice of δ thus translates into a choice of selecting m ∈ {0, 1, 2, ..., N } such that the firm’s expected profits are
maximized when the retailers with the m lowest MADs participate in the scheme.11 Unfortunately, for an unrestricted
10
There is a large literature on supply chain coordination via return contracts (see Pasternack 1985, Donohue 2000,
Taylor 2001). In those settings, the return policy is adopted to mitigate retail risk of overstock , thereby incentivizing
retailers to order the supply chain optimal quantity. There is a certain range of return prices, closely related to the
wholesale price, that achieve this goal. We remark that a consumer return policy plays an important role in resource
allocation, and the optimal return price maximizes social welfare [Su, 2009]. In our model, however, the return policy
is used to mitigate the supply risk of understock , and the supply chain is coordinated under any value of return price
v > 0. This allows us to separate the decision of v and δ and to develop our results for an exogenously specified value
of v > 0.
11
If m = 0 then δ = 0; that is, the firm does not offer any discount.
42
WENJIE TANG AND KARAN GIROTRA
set of retailer characteristics (µi , σi , ti , Ki ), the firm’s expected profits are not monotonic in m; hence there is no
closed-form solution to the choice of m.12 The algorithm to choose m is relatively simple, though. First, the firm sorts
the retailers in the order of their MADs (Equation 7.1). Next, the firm compares its expected profits for different
choices of m, where the expected profits are computed by using Equation (7.2) and assuming the retailers with the
m lowest MADs will participate.
As Equation (7.2) illustrates, the firm’s expected profits depend on each participating retailer’s characteristics: the
prior demand distribution N (µi , σi2 ) faced by the retailer, her private information quality ti , and her administrative
cost Ki , each of which has a different effect on the firm’s expected profits. This is why we do not provide, beyond the
algorithm already proposed, a general conclusion or a structural property that drives the final choice. In the following
analysis, we assume that the firm has already applied that algorithm and has chosen the m > 0 desirable retailers.
Our next lemma formally captures the changes just described in the firm’s strategy and explains the intuition behind
the second part of the firm’s strategy profile: the order quantity.
Lemma. Firm’s Strategy Profile: Suppose the firm offers the APD scheme to N retailers with MADs δ (1) ≤
δ (2) ≤ · · · · · · ≤ δ (m) ≤ · · · · · · ≤ δ (N ) , where 1 ≤ m ≤ N and, for all i, δ i is defined as Equation (7.1). Then the firm
0
0
sets the discount δ = δ (m) , to make the first m retailers participate, and places an overseas order P = µ + σ zγ . The
resulting expected profits become
Eπ F
=
m h
X
0
((1 − δ) w − (1 − γ)) µi + (w − v) φ (zδ ) σi
i
i=1
+
(7.2)
N
X
0
(w − (1 − γ) CL ) µi − CL φ (zγ ) σ ,
i=m+1
0
where µ =
Pm
i=1
0
µi +
PN
i=m+1
0
µi , σ =
qP
m
i=1
0
σi 2 +
PN
i=m+1
σi2 , zδ = Φ−1
wδ
w−v
, and zγ = Φ−1 (γ).
Instead of ordering a unique Pi for each retailer, now the firm considers the demand from all N retailers as a whole
and orders P units for all of them. The total demand consists of two parts: for participating retailers, the firm infers
demand signals from their order quantities and updates its posterior distribution of total demand for m retailers to
P
PN
PN
0 Pm
0
2
2
N( m
i=1 µi ,
i=1 σi ); for nonparticipating retailers, the posterior distribution remains N (
i=m+1 µi ,
i=m+1 σi ).
P
0
m
Therefore, the total demand faced by the firm has a posterior normal distribution with mean µ0 =
i=1 µi +
q
PN
Pm 0 2 PN
0
2
i=m+1 µi , and standard deviation σ =
i=1 σi +
i=m+1 σi . The firm’s optimal order quantity and expected
profits follow directly from the same newsvendor problem under customized products as in Section 3 but with the
new demand distribution.
In the case of generic products, the firm offers the same discount to all N retailers instead of setting a separate
discount at each retailer’s MAD; this offered discount is equal to the highest MAD among all m retailers. Thus, in
equilibrium, retailers with the m lowest MADs would participate in APD and the rest would be better off staying
out. The following theorem formally describes the equilibrium.
12
It is relatively easy to derive closed-form results when all retailers are identical or by restricting retailer heterogeneity
in other ways. Yet because we feel this has limited practical appeal, the most general model is reported here.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
43
Theorem 7. Equilibrium Outcome
(1) Actions: The tuple of actions {δ ∗ , Q∗i , P ∗ } characterizes a perfect Bayesian equilibria of the setup just
described. For all i,
δ ∗ = δ (m) ,
0
0
Q∗i = 1i≤m µi + σi zδ∗ ,
0
0
P ∗ = µ + σ zγ .
(2) Profits: The equilibrium expected profits for retailer i are EΠR
i , and the equilibrium expected profits for the
firm EΠF are
P
F
i=1..N EΠi ,
where
EΠR
i
=
h
i
0
1i≤m (Ri − (1 − δ ∗ ) w) µi − (w − v) φ (zδ∗ ) σi − Ki + 1m<i≤N [(Ri − w) µi ] ,
EΠF
=
m h
N
i
X
X
0
((1 − δ) w − (1 − γ)) µi + (w − v) φ (zδ ) σi +
(w − (1 − γ) CL ) µi
i=1
i=m+1
0
−CL φ (zγ ) σ .
We next examine the supply chain performance under optimal policy.
Theorem 8. Benefits of Advance Purchase Discounts: With an APD scheme using δ ∗ as defined in Theorem
7, participating retailers, the firm, and the supply chain are all better off. The benefits are given as follows:
h
i
0
∗
∆EΠR
i = 1i≤m δ wµi − (w − v) φ (zδ ∗ ) σi − Ki ≥ 0,
∆EΠF =
m h
X
i
0
−δ ∗ wµi + (w − v) φ (zδ∗ ) σi + K̄ ≥ 0,
i=1
∆EΠS = K̄ − K ≥ 0,
where K̄ = CL φ (zγ )
P
N
i=1
σi − σ
0
and K =
Pm
i=1
Ki .
Although with generic products the firm adopts a different discount strategy, the description of the APD benefits in
Theorem 8 is the equivalent of Theorem 1 for customized products. In fact, the intuition follows along exactly the
same lines as before.
To participate in the APD scheme, each retailer incurs an additional administrative cost Ki (Effect Admin). By
offering the APD to retailers, the average transfer price between the firm and each participating retailer is reduced
by δ ∗ wµi , which leads to higher expected profits for each retailer but to lower ones for the firm (Effect Discount). At
0
this cost, the firm invites each participating retailer into sharing demand uncertainty (w − v) φ (zδ∗ ) σi (Effect Risk).
Furthermore, under the APD scheme, superior demand information flows from the downstream tier (the retailers) to
the upstream tier (the firm); hence the firm derives benefits K̄ from all participating retailers by using is information
to make sourcing decisions that utilize long–lead time suppliers (Effect Information). Despite the internal effects
(Effect Discount and Effect Risk), the supply chain benefits K̄ from superior information even though it comes with
extra administrative costs K. As before, the combination of these effects drives the benefits for all agents in the supply
chain. However, there is now one notable difference. With customized products, we recall, the firm is able to tailor
44
WENJIE TANG AND KARAN GIROTRA
a different discount for each retailer and thus can appropriate all supply chain benefits. Yet with generic products,
the firm offers all N retailers the same discount δ (m) , which is the lowest MAD among m retailers. Therefore, while
retailers with higher MADs would have no incentive to participate, retailers with lower MADs can obtain strictly
positive benefits from participating in the APD scheme. This makes all the participating agents, except retailer m,
strictly better off under Advance Purchase Discounts.
8. Proofs of Results in Preceeding Sections
8.1. Proofs of Results in Section 3.
Proof of Lemma: Retailers’ Strategy Profile
Proof. Given δi , retailer i’s profit if participating in the APD is πiR = Ri Di − (1 − δi ) wi Qi − wi (Di − Qi )+ − Ki ,
0
0
which has a typical newsvendor solution with optimal order quantity Qi = µi + σi zδi and corresponding expected
0
profits EπiR = (Ri − (1 − δi ) wi ) µi − wi φ (zδi ) σi − Ki , where zδi = Φ−1 (δi ).13 On the other hand, recall that retailer
R
= (Ri − wi ) µi . The two expected profits are equal
i’s expected profits when not participating in the APD are Eπi0
n o
0
when δi = δ i , where δ i = δi δi wi µi − wi φ (zδi ) σi − Ki = 0 . The difference between the two expected profits is
0
φ(zδ ) 0
R
R
i
Eπi − Eπi0 = δi wi µi − wi φ (zδi ) σi − Ki = wi Φ (zδi ) µi − Φ z σi − Ki (recall that δi = Φ (zδi )). The standard
( δi )
φ( z )
normal distribution is log-concave and thus has decreasing reversed hazard rate (DRHR); that is, Φ zδi is decreasing
( δi )
R
is nonnegative and increasing in zδi (and hence increasing in δi ); if
in zδi . Therefore, if δi ≥ δ i , then EπiR − Eπi0
R
δi ≤ δ i , then EπiR < Eπi0
and retailer i does not participate in the offered APD scheme. This concludes the proof.
Proof of Lemma: Firm’s Strategy Profile
Proof. Given δi and retailer i’s participation, the firm’s profit is πiF = (1 − δi ) wi Qi +wi (Di − Qi )+ −(1 − γ) CL Pi −
0
CL (Di − Pi )+ . This expression, which has a typical newsvendor solution with optimal order quantity Pi = µi +
0
0
0
σi zγ and corresponding expected profits EπiF = ((1 − δi ) wi − (1 − γ) CL ) µi + wi φ (zδi ) σi − CL φ (zγ ) σi for re0
0
tailer i’s corresponding order quantity Qi = µi + σi zδi , where zγ = Φ−1 (γ) and zδi = Φ−1 (δi ). On the other
F
hand, recall that the firm’s expected profits when retailer i does not participate in the APD scheme are Eπi0
=
F
(wi − (1 − γ) CL ) µi − CL φ (zγ ) σi . The difference between the two expected profits is EπiF − Eπi0
= −δi wi µi +
0
0
wi φ (zδi ) σi + CL φ (zγ ) σi − σi , which from the previous proof, is decreasing in δi when δi ≥ δ i . To ensure retailer
0
F
i’s participation, the firm’s optimal discount is therefore δi = δ i , which makes EπiF −Eπi0
= −Ki +CL φ (zγ ) σi − σi .
0
Thus, the firm would offer an Advance Purchase Discount to retailer i if and only if −Ki + CL φ (zγ ) σi − σi ≥ 0.
This concludes the proof.
13
Unless specifically indicated otherwise, all expectations of profits in this paper include expectation over both
demand and available private signal; all expectations of benefits include expectation over demand, private signal, and
administrative cost. Benefits alone refer to the difference between expected profits with and without APDs.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
45
Proof of Theorem 1
Proof. The benefits that accrue to retailer i, the firm, and the supply chain under Advance Purchase Discounts
stem directly from the difference between the expected profits with and without the APD scheme. For δi ≥ δ i , the
firm’s expected profits are decreasing in δi ; therefore, the maximum discount it could offer is the δ̄i that yields the
o
n 0
firm zero benefits from offering APD scheme—that is, when ∆EπiF = 0 for δ̄i = δi δi wi µi − wi φ (zδi ) σi − K̄i = 0 .
0
Comparing the definition of δ i and δ̄i : if Ki < K̄i , then δ i < δ̄i and Ki < δi wi µi − wi φ (zδi ) σi < K̄i for a δi ∈ δ i , δ̄i .
Proof of Theorem 2
Proof. In equilibrium, when it is beneficial to employ the APD scheme (i.e., when Ki ≤ K̄i ), the firm offers a
discount that simultaneously maximizes its expected profits and ensures retailer i’s participation (i.e., δi∗ = δ i ); in
the second stage, each agent orders the newsvendor optimal quantity. In contrast, when it is not beneficial to employ
the APD scheme (i.e., when Ki > K̄i ), the firm offers no discount. This concludes the proof.
8.2. Proofs of Results in Section 4.
Proof of Lemma
Proof. The firm’s expected benefits from offering the APD scheme with a discount δi are
E∆π̂iF



0,




= (1 − b) −δi wi µi + wi φ (zδi ) σi0 + CL φ (zγ ) σi − σi0
,





0
0

−δi wi µi + wi φ (zδi ) σi + CL φ (zγ ) σi − σi ,
δi < δ i2 ,
δ i2 ≤ δi < δ i1 ,
δi ≥ δ i1 .
For δ i2 ≤ δi < δ i1 and δi ≥ δ i1 , E∆π̂iF is decreasing in δi ; hence the optimal discount reduces to either δ i1 or δ i2 .
The lemma follows from comparing
0
0
(1 − b) −δ i2 wi µi + wi φ zδi2 σi + CL φ (zγ ) σi − σi
and
0
0
−δ i1 wi µi + wi φ zδi1 σi + CL φ (zγ ) σi − σi .
Proof of Theorem 3
Proof. First we consider the optimal discount to set if the firm offers the APD scheme. (i) Bi ≤ K̄i . Recall that
0
K̄i = CL φ (zγ ) σi − σi . In response to an offer of discount δi , retailer i would participate in the APD if and only if
46
WENJIE TANG AND KARAN GIROTRA
0
she would have nonnegative benefits under this discount (i.e., when δi wi µi − wi φ (zδi ) σi ≥ Ki ). Taking into account
retailer i’s participation, the firm’s expected benefits are
h
i 0
0
EKi ∆Eπ̂iF (δi ) = −δi wi µi + wi φ (zδi ) σi + K̄i Gi δi wi µi − wi φ (zδi ) σi .
0
Define ki = δi wi µi − wi φ (zδi ) σi , EKi ∆Eπ̂iF (δi (ki )) = K̄i − ki Gi (ki ), and
!
∂EKi ∆Eπ̂iF (δi (ki ))
K̄i − ki gi (ki )
= −Gi (ki ) 1 −
,
∂ki
Gi (ki )
∂E
[∆Eπ̂F (δi (ki ))]
gi (ki )
where ki ∈ 0, K̄i . Since gi (·) has a DRHR, it follows that G
decreases in ki and that Ki ∂kii
decreases
(k
)
i
i
F
g
(k
)
g
(k
)
K̄
−k
(δ
(k
))
K̄
−K
∂E
∆Eπ̂
( i i) i i
( i i) i i
[
i i ]
≥ 0. Let ki0 = ki = 1 . For ki ∈ 0, K̄i , Ki ∂kii
is positive
in ki when 1 −
Gi (ki )
Gi (ki )
∗
when ki ∈ [0, ki0 ] but is decreasing and negative when ki ∈ ki0 , K̄i . Therefore, if Ai ≤ ki0 ≤ Bi then ki = ki0 ; if
ki0 < Ai then ki∗ = Ai ; and if ki0 > Bi then ki∗ = Bi . (ii) Bi > K̄i , the firm’s expected benefits become14
h
i 0
0
EKi ∆Eπ̂iF (δi ) = −δi wi µi + wi φ (zδi ) σi + K̄i Gi δi wi µi − wi φ (zδi ) σi Gi K̄i ,
which under the same transformation becomes EKi ∆Eπ̂iF (δi (ki )) = K̄i − ki Gi (ki ) Gi K̄i . Therefore, if Ai ≤
ki0 ≤ K̄i then ki∗ = ki0 ; if ki0 < Ai then ki∗ = Ai .
Next, we consider the firm’s decision to offer the APD scheme or not. By offering the APD scheme with discount
δ̂i k̂i , the firm’s expected benefits become K̄i − k̂i Gi k̂i ; therefore, the firm would only offer the APD scheme
when it gets nonnegative expected benefits, i.e., when k̂i ≤ K̄i . This concludes the proof.
Proof for Theorem 4
Proof. Observe that gi (ki |ui , si ) =
1
g
si i0
ki −ui
si
1
si
(8.1)
When si = si0 =
(K̄i −ui )gi0 (0)
Gi0 (0)
. By Equation (4.2),
i
K̄i − ki0 gi0 ki0s−u
i
= 1.
i
Gi0 ki0s−u
i
, we have ki0 = ui . First we show that, for a given ui , if si < si0 then ki0 > ui and
therefore δ̂i (ki0 ) > δ̂i (ui ). Suppose to the contrary that si < si0 but ki0 ≤ ui . Then, by Equation (8.1), we must
1 K̄ −k
i
( i i0 )gi0 ki0s−u
i
have si
= 1. However, the DRHR of gi0 (·) yields that
ki0 −ui
Gi0
si
1
si
i
K̄i − ki0 gi0 ki0s−u
i
>
i
Gi0 ki0s−u
i
1
si0
K̄i − ui gi0 (0)
Gi0 (0)
= 1,
a contradiction. Therefore, if si < si0 then ki0 > ui . Similarly, suppose that si > si0 but ki > ui . Then
1
i
1
K̄i − ki0 gi0 ki0s−u
K̄i − ui gi0 (0)
si
i
si0
<
= 1,
Gi0 (0)
i
Gi0 ki0s−u
i
contradicting Equation (8.1). Therefore, if si > si0 then ki0 < ui . This concludes the proof.
14
If Ki > K̄i then the APD is not offered and so the firm’s benefits are zero.
SYNCHRONIZING GLOBAL SUPPLY CHAINS: ADVANCE PURCHASE DISCOUNTS
47
Proof of Theorem 5
Proof. By Theorem 3, the firm would treat retailer i as if her administrative cost were k̂i . If k̂i > K̄i and k̂i ≤ K̄i ,
it would not be beneficial for the firm to offer the APD scheme so δ̂i∗ = 0. If k̂i < Ki , it would be beneficial for
the firm to offer the APD scheme and so δ̂i∗ = δ̂i k̂i ; however, in this case it would not be beneficial for retailer i
to participate. In both cases, the APD scheme cannot be implemented. In contrast, when Ki ≤ k̂i ≤ K̄i , the firm
would offer the APD scheme with discount δ̂i k̂i and retailer i would participate. The optimal order quantities and
expected profits thus follow.
Proof of Theorem 6
Proof. The benefits follow directly from the difference between the two expected profits in Theorem 5, bearing in
0
mind that K̄i = CL φ (zγ ) σi − σi .
Proof of Corollary
Proof. The corollary follows from the benefits listed in Theorem 6 and the statements in Theorem 4 that k̂i (ui , si ) ≥
ui when si ≤ si0 and that k̂i (ui , si ) < ui when si > si0 .
8.3. Proofs of Results in Section 7.
Proof of Lemma: Retailers’ Strategy Profile
Proof. Given δ, retailer i’s profit if participating in the APD is πiR = RDi − (1 − δ) wQi − w (Di − Qi )+ +
0
0
v (Qi − Di )+ − Ki , which has a typical newsvendor solution with optimal order quantity Qi = µi + σi zδ and cor
0
wδ
. On the
responding expected profits EπiR = (R − (1 − δ) w) µi − (w − v) φ (zδ ) σi − Ki , where zδ = Φ−1 w−v
R
other hand, recall that retailer i’s expected profits if not participating in the APD are Eπi0
= (R − w) µi . The two
o
n 0
expected profits are equal when δ = δ i , where δ i = δ δwµi − (w − v) φ (zδ ) σi − Ki = 0 . The difference between
0
φ(zδ ) 0
R
the two expected profits is EπiR − Eπi0
= δwµi − (w − v) φ (zδ ) σi − Ki = (w − v) Φ (zδ ) µi − Φ(z
σi − Ki ; recall
δ)
that δ =
(w−v)Φ(zδ )
.
w
The standard normal distribution is log-concave and thus has DRHR; in other words,
decreasing in zδ . Therefore, if δ ≥ δ i then
EπiR
−
R
Eπi0
φ(zδ )
Φ(zδ )
is
is nonnegative and increasing in zδ (and hence increasing in
R
δ); if δ ≤ δ i then EπiR < Eπi0
and retailer i does not avail herself of the offered Advance Purchase Discount This
concludes the proof.
Proof of Lemma: Firm’s Strategy Profile
48
WENJIE TANG AND KARAN GIROTRA
Proof. Given δ and the participation of m retailers, the firm’s profit is
πF
=
(1 − δ) w
m
X
Qi + w
m
X
(Di − Qi )+ − v
i=1
i=1
− (1 − γ) CL P − CL
m
X
+
(Di − Qi ) +
=
−δw
Qi + (w − v)
i=1
m
X
m
X
(Qi − Di ) + w
i=1
Di
i=m+1
Qi +
i=1
+
N
X
(Qi − Di )+ + w
i=1
i=1
m
X
m
X
N
X
N
X
i=m+1
Di −
m
X
!+
+
(Qi − Di ) − P
i=1
Di − (1 − γ) CL P − CL
i=1
N
X
!+
Di − P
.
i=1
0
0
This expression has a typical newsvendor solution with optimal order quantity P = µ + σ zγ and corresponding
expected profits
Eπ F =
m h
X
N
i
X
0
0
((1 − δ) w − (1 − γ)) µi + (w − v) φ (zδ ) σi +
(w − (1 − γ) CL ) µi − CL φ (zγ ) σ ,
i=1
i=m+1
P
PN
0
0
0
0
0
given each participating retailer i’s corresponding order quantity Qi = µi + σi zδ for µ = m
i=1 µi +
i=m+1 µi , σ =
qP
PN
0
02
m
−1
wδ
2
, and zγ = Φ−1 (γ). Since (1 − δ) w + (w − v) φ (zδ ) σi is decreasing in
i=1 σi +
i=m+1 σi , zδ = Φ
w−v
δ when δ ≥ δ i , it follows that Eπ F is decreasing in δ when δ ≥ δ (m) . To maximize the firm’s expected profits Eπ F
and to ensure the first m retailers’ participation, the optimal discount for the firm becomes δ (m) . This concludes the
proof.
Proof of Theorem 7
Proof. In equilibrium, when it is beneficial to offer the APD scheme to the first m retailers, the firm offers a discount
that simultaneously maximizes its expected profits and ensures the participation of those retailers (i.e., δ ∗ = δ (m) ). In
the second stage, each agent orders the newsvendor optimal quantity, and the corresponding expected profits follow.
Proof of Theorem 8
Proof. Recall that, when no APD is implemented, the respective expected profits of retailer i, the firm, and the supply
R
chain are Eπi0
= (R − w) µi , Eπ0F =
PN
i=1
((w − (1 − γ) CL ) µi − CL φ (zγ ) σi ), and Eπ0S =
PN
i=1
((R − (1 − γ) CL ) µi − CL φ (zγ ) σi ).
The benefits of APDs are a direct consequence of the difference between the expected profits in Theorem 7 and the
corresponding expected profits just described.